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An Intuitive Guide to Linear Algebra (2012) 709 points by PNWChris 34 days ago | hide | past | web | favorite | 102 comments

 I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head.Now that the ideas of things like vector spaces, norms, orthogonality, rank, basis, etc are nearly second nature, the concepts are useful as I study other branches of math which would feel impenetrable otherwise.YMMV, and if you can learn from condensed materials go for it, but I might be too dumb for it work lol. I think the real benefit accrues to the author who had to work out how to teach these concepts to others.
 My experience from a decade of doing professional maths is that there are no shortcuts. You only learn maths by doing hundreds (thousands) of exercises, both mundane and more exctiting.Also, the concepts "mature" in the brain. I remember sleepless nights in the first year of undergrad spent on understanding the details of the proof of the Jordan decomposition and a few years later (when studying algebraic groups) it all felt trivial.There's no shortcut to understanding maths, just a lot of time spent in solitude trying to make sense of all these abstract concepts (and they DO make sense).
 I think there is a shortcut: use examples (a "domain") rather than starting with theory. learning something purely in the abstract is easy to forget.
 I agree, and have to admit that my own knowledge of LA is sadly way too superficial. The article did give a me a big lightbulb on something I didn't understand before: Some of my friends work in quantitative sociology and economics, and use Stata for their programming needs. They basically use matrices as their exclusive data structure, and the reason for that eluded me until after reading this article.
 > While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head.I'm not sure I understand your point. Are you just saying that this blog post isn't an adequate substitute for taking a course in linear algebra? (Of course it isn't. But who said it was?)
 Unfortunately, for a lot of people, including undergraduates, the dopamine hit they get from watching a video or passively reading a textbook makes them believe that these are adequate substitutes for doing thousands of exercises.In my university, undergraduates have admitted that they have done fewer than 50 questions throughout the entirety of my math course. Their grades obviously reflect that, but they will do the same next semester.
 Can I press you to explain what you think dopamine is/does?I'm curious because I'm a neuroscientist who occasionally works on dopaminergic stuff and "passively reading a textbook" is so far from the canonical examples we use for dopamine activity, but the idea of dopamine/dopamine 'hits' has taken on a life of its own that seems quite different from the neurotransmitter.
 I am a physicist, so I will dare not make any technical neuroscience claims. I meant "dopamine" in the causal sense of people feeling pleasure from passively reading a book because they think it is useful work.
 There is a lot of excitement around machine learning and AI and there is not a proportional excitement for the math that underlies the theory.In a lot of content that teaches machine learning/AI, the linear algebra substrate of it is given short shrift.The consumers of that content infer that the backing mathematics is easy or unimportant, while the creators of the content are not actually implying that, but just want to move on to what the audience came for.I'm not an expert in machine learning so I can't say whether you can get by without a strong understanding of linear algebra, but my intuition answer is no you can't. Beside that, I enjoy the math for its own sake so I'm happy to trudge through the textbooks.And no, I don't think the blog post makes any claims that it is a subsitute for taking a course in linear algebra.
 3blue1brown himself has told in his videos that they are not substitute for books and working out the exercise and he recommends reading from books and his videos are for inspiration and as a supplement.
 > I think the real benefit accrues to the author who had to work out how to teach these concepts to others.In college I learned more math when I was trying to build software for teaching math compared to when I was trying to learn math.
 Yeah, this sort of stuff is fine supplementary material... but nobody really learns anything from a blog post or some videos.
 I disagree. Maybe you'll remember it not as well if you don't do exercises, but there is no reason you can't learn from a blogpost (which is trivial to prove since you can just copy-paste the contents of a book in the form of one or more blogposts).
 What your parent comment meant was that it is not possible to learn well from typical blog posts like this that try to condense the subject into a 3000 word article. Of course, if you copy-paste the content of a book into a blog post, then parent comment's point no longer applies.
 I'd say it still applies, even when reading a textbook going through the exercises is crucial
 While this explanation is certainly much clearer than what I remember of high school maths, I still have a pretty tough time following the formula examples.When I see A(x) = ax, I'm not entirely sure how to read it.Is A meant to be a function that accepts x? If so, why is the equivalent expression a * x? Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by the supplied value? Is this notation specific to multiplication, to this expression, or what?Positing that something is 'intuitive' when it depends so much on additional contextual knowledge seems ever so slightly disingenuous as best, and slightly harmful at worst; it can make the reader feel as though they must be dumb for not understanding this 'intuitive' material.I do acknowledge that this is linear algebra, and if one doesn't have a really solid grasp of notation of regular algebra it is likely to go over their heads, but the practical explanations (such as the slope rise/run example) are quite clear and relatively simple to follow; it follows that a simple explanation of the notation might be helpful too.
 I agree that this was a bit confusing. Higher up in the article, it shows that a linear function is one that doesn't change when scaled:F(a * x) = a * F(x)This is showing the relationship between two uses of the same function.Then, further along, we find:"So, what types of functions are actually linear? Plain-old scaling by a constant, or functions that look like: F(x)=ax In our roof example, a=1/3"I think in this second situation, F(x)=ax is not a relationship but rather a DEFINITION of the function F(x).In programming terms:function F(x: real) : real;begin Result := x * (1/3);  end;
 The particular example might be confusing because the same letter is used twice with different capitalization. There is no direct relation between them.> Is A meant to be a function that accepts x?Yes.> If so, why is the equivalent expression a * x?Because that how A(x) is defined.>Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by the supplied value?Yes, it's an unspecified constant (a, b, c... are used to denote constants by convention), so you can really calculate A(x) for supplied values of x yet until the constant 'a' is specified.> Is this notation specific to multiplication, to this expression, or what?No. Functions might be defined using any expression. For example A(x) = b^x  is a valid function as well (again, we have an unspecified constant). Just don't expect to encounter it in an introductory course in linear algebra (since it would deal mostly with linear functions).
 This is ok but nothing is as intuitive as 3B1B's series on YouTube that has been posted hundreds of times on HN [0].Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post.
 > Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post.I... disagree. Some of linear algebra is about that. And it's probably a good way to view it that way when learning.But some of my current work (coding theory) involves linear algebra over finite fields. We use results from linear algebra, and interpret our problem using matrices, but really at no point are we viewing what we're doing as transforming a vector space, we're just solving equations with unknowns.
 I think this is spot on. Depending on what you're doing, a matrix can be: - A linear transformation - A basis set of column vectors - A set of equations (rows) to be solved - (your example: parity equations for coding theory) - The covariance of elements in a vector space - The Hessian of a function for numerical optimization - The adjacency representation of a graph - Just a 2D image (compression algorithms) ... (I'm sure there are plenty of others)  For some of these, the matrix is really just a high dimensional number. You (rarely?) never think of covariance in a Kalman filter as a linear transform, but you still need to take its Eigen vectors if you want to draw ellipses.
 The first three can reasonably be thought of as defining linear transformations. For linear systems of equations A x = b, x is an unknown vector in the input space that is mapped by A to b.Both covariance matrices and Hessians are more naturally thought of as tensors, not matrices (and therefore not linear transformations). That is, they take in two vectors as input and produce a single real number as output.As for graph adjacency matrix, this can actually be thought of as a linear transformation on the vector space where the basis vectors correspond to nodes in the graph. Linear combinations of these basis vectors correspond to probability distributions over the graph (if properly normalized).2D images... Yes, these cannot really be interpreted as linear transformations. But I'd say these aren't really matrices in the mathematical sense.
 If you squint hard enough, you can see all of them as linear transformations (even the 2D images :-).I politely disagree about covariance and Hessians. I can squint and say that the Hessian provides a change in gradient when multiplied by a delta vector. Similarly for covariance... Or you could look at it as one half of the dot product for a Bhattacharyya distance, which is just a product of three matrices (row vector, square matrix, col vector). No need for tensors yet.That is unless you decide to squint hard enough to see everything as tensors! :-)
 Great points. I wrote my comment in response to the article claiming to be an intuitive guide to linear algebra, not an intuitive guide to matrices. According to wikipedia:> Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces through matrices. [0]The Venn Diagram of Linear Algebra and Matrices definitely has a lot of non-overlap, of which your list covers some. This article should be renamed to be about matrices and not linear algebra, because it's not.
 A covariance matrix naturally transforms from the measured space to a space where things are approximately unit Gaussian distributed. This is identical to the Z transform in 1D case.This can be useful in, say, exotic options trading - a natural unit of measurement is how many ‘vols’ an underlier has moved, e.g. a 10-vol move is very large.
 Not really the covariance matrix, though, but its Cholesky decomposition (which exists, as a covariance matrix is symmetric positive (semi)definite, as otherwise you could construct a linear combination with negative variance). Useful stuff.And vice versa, btw - take iid RV with unit variance, hit them with the Cholesky decomposition, and you have the desired covariance. Used all over Monte Carlo and finance and so on.
 > - A basis set of column vectorsLet's leave the word 'basis' out, since the column vectors may well be linearly dependent.
 Or not span the space, for that matter.
 Well, it depends on what "the space" is. Every set of vectors (in a common ambient space) spans some space—often called the column space of a matrix, if they are the column vectors of the matrix.
 Sure, every set of vectors will span the space they span. But the requirement that a basis span the space refers to the space it’s in, not the space it spans (otherwise every linearly independent set of vectors were a basis, spanning the space it spans.) I could go on :-)
 > > Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post.> I... disagree.This is literally the definition of the term "linear algebra".> really at no point are we viewing what we're doing as transforming a vector space, we're just solving equations with unknowns.You may not see what you're doing as transforming vector spaces with linear operators, but that is what you're doing. It's worth pointing out that the definition of vector spaces allows any field, including finite ones, though it's true that the intuition won't be exactly the same.Another way to say this: if you're working on a problem without thinking about the connection to linear transformations, then it's not correct to say it's a linear algebra problem without obvious connection to linear transformations; instead, it's not a linear algebra problem at all, by definition.
 Linear algebra is a shared field across multiple disciplines. So I'm sure that there are many valid and useful interpretations as to what "linear algebra" is essentially about.However, in mathematics proper, it is absolutely the case that linear algebra is about linear transformations. Indeed, this is the only interpretation that remains meaningful when trying to generalize (e.g. to functional analysis / multilinear algebra).
 20+ years ago I took a grad course in coding theory, e.g.,W. Wesley Peterson and E. J. Weldon, Jr., Error-Correcting Codes, Second Edition, The MIT Press.-- gee, people are still studying/learning that?The prof knew the material really well, but to up my game in the finite field theory from other courses, I usedOscar Zariski and Pierre Samuel, Commutative Algebra, Volume I, Van Nostand, Princeton.which did have a lot more than I needed!My 50,000 foot overview of linear algebra is that the subject still rests on the apparently very old problem of the numerical solution of systems of simultaneous (same unknowns) linear equations, e.g., via Gauss elimination (it's really easy, intuitive, powerful, and clever, surprisingly stable numerically, and is fast and easy to program; someone might want to type in, say, just an English language description!). Since such the subject of linear equations significantly pre-dates matrix theory, the start of matrix theory was maybe just easier notation for working with systems of linear equations. In principle, everything done with matrix theory could have been with just systems of linear equations although often at a price of a mess notationally. In particular, as I outline below, now there are lots of generalizations of systems of linear equations that use different notation and not much matrix theory.What's amazing are the generalizations, all the way to linear systems (e.g., their ringing) in mechanical engineering, radio astronomy, molecular spectroscopy, frequencies in radio broadcasting, stochastic processes, music, mixing animal feed, linear programming, oil refinery operation optimization, min-cost network flows, non-linear optimization, Fourier theory, Banach space, oil prospecting, phased array sonar, radar, and radio astronomy, seismology, quantum mechanics, yes, error correcting codes, linear ordinary and partial differential equations, ..., and thenNelson Dunford and Jacob T. Schwartz, Linear Operators Part I: General Theory, ISBN 0-470-22605-6, Interscience, New York.
 Linear algebra IS about linear transformations and vector spaces.The thing is that the field over which the space is defined can be quite arbitrary (finite, infinite, not algebraically closed etc.) which has immense consequences on the behavior of such objects.When one drops the assumption on finite number of dimensions, the story becomes wild (and is known as functional analysis, beautiful and extremely useful branch of mathematics).
 3B1B does great work explaining these concepts, but I can't help but ask "why not both?" when it comes to explaining these concepts. Turns out, linear algebra is great for working with matrices, vector space, approximating non-linear systems, and more... Let's embrace multiple ways of teaching it and gaining intuitions rather than keeping score, eh?
 Then you failed to comprehend the subject. The point is that a wide array of problems and models are really the same thing.
 Failed to comprehend which subject, linear algebra? I would argue no, and other people who are more on the pure mathematics side would agree [0][1].snicker7 said it very succinctly:> However, in mathematics proper, it is absolutely the case that linear algebra is about linear transformations. Indeed, this is the only interpretation that remains meaningful when trying to generalize (e.g. to functional analysis / multilinear algebra).If you're point is that I failed to comprehend matrices, then I don't think you have enough data to make that claim since I don't really talk about matrices. I kind of address that in my other comment [2].I don't follow your point around "a wide array of problems and models are the same thing". That's a very vague general statement that I certainly comprehend (not sure how you inferred otherwise). Specifically, I don't see how that point relates at all to the claim I made about linear algebra.
 I... I still really struggle with this. I'm a smart person, I've got a bachelors of engineering, I've been a professional software developer for around 14 years now, and I've built a house. But there is something about degree-level maths and beyond that I find deeply unintuitive in a way that software development isn't.Through comments here I found 3blue1brown's (clearly much loved) videos. By the third video I was shouting, "why for the love of god would we be doing this"? Based on this reaction I suspect that the content neither has intrinsic appeal to me, nor does it have obvious use in my work, projects, or life.Pre-degree maths though, I love. My A-level maths really changed how I saw the world, and I make use of it reasonably often (well, often enough to not forget it).I think I'm writing this here because most other commenters seem to really grasp this subject, or feel that they grasp it better having seen these videos. I'm honestly happy for you. However, if anyone is reading this who doesn't feel like that, then know you're not alone :-)
 > But there is something about degree-level maths and beyond that I find deeply unintuitive in a way that software development isn't.Grok.Over the years, I come to the conclusion that one of the stumbling blocks is the definition/concept of "application". Just like the definition of "theory" is different for a layman ("My theory is..." == "My guess is...") than from a scientist's definition ("My theory is" == "My logical framework which incorporates all of the available data is..."), so the definition of "application" is different between mathematicians and engineers.I've noticed that math books with titles like "$HIGHER_ORDER_MATH with Applications" means "$HIGHER_ORDER_MATH with Exercises". What I'm looking for is something like "\$HIGHER_ORDER_MATH with Real-World Uses".I've known LA for decades but, like you, where would I use it in my life? The turning point for me was Andrew Ng's Deep Neural Network course.I knew that a DNN is a program of matrix operations, but how do you get 5,000 images into a matrix? One way is to resize all the images to the same n x n size, take the first pixel of each picture and break them into their RGB components. You now have the first three rows of your input matrix. Repeat for all other pixels and voila! You have a 5,000 x n matrix that you can do linear algebra on! _That's_ an application; having me add two matrices together is an exercise.Since that insight, I've used LA in my job in the hospitably sector with impressive success because now I know how to apply it. Math books and 3B1B show you the math. We engineers (or at least this one) need real world uses.
 This is typical for many people. You love pre-college math because you have intuitive understanding, while college-level maths offer a new level of abstraction that you may not feel familiar with from the get-go.It's perfectly okay not to learn linear algebra, by the way, especially when you don't find any incentive to do so. Otherwise, you'll find linear algebra to be one of the most intuitive tools to model so many problems.If you do want to learn linear algebra or any other higher math, I'd strongly recommend you focus on understanding concepts intuitively first, to the point that you find many exercises in a text book straight forward. Watching 3blue1brown is a good start, but do move forward with deeper treatment. The book I find very usual is David Lay's Linear Algebra and Its Applications: https://www.amazon.com/Linear-Algebra-Its-Applications-5th/d.... Lay sets up a really intuitive geometric framework to explain the intuition of linear transformation with sufficient rigor.
 Got any similar recommendations for college-level probability/stats stuff?
 No. I took only a few courses on probability and mathematical stats. The books are A First Course in Probability and some some textbook on Mathematical Stats. The courses were probably not advanced enough, as I found them reasonably straightforward. I struggled a bit on what exactly is a random variable, but once that was internalized, everything else followed. I heard that advanced courses like random processes were really hard, but I'm not at that level.
 Part of the challenge with linear algebra is that a lot of the basics are somewhat dry, since they serve mostly as a way to organize computation -- e.g., a system of linear equations can be expressed as Ax <= b, where A is a matrix, b is a vector, and x is a vector of variables.Much of what makes linear algebra interesting and powerful comes from more advanced topics, especially eigenvalues. This power comes when we are not looking at a single matrix in isolation, but when we repeatedly apply a matrix. For instance, consider the equation x_t = A^t x_0. It turns out we can rewrite this an equation by diagonalizing A -- i.e., A = P^{-1} Sigma P, where Sigma is diagonal; most, but not all, matrices can be written in this form. We call s_i the "eigenvalues" of A.Then, the equation simplifies to x_t = P^{-1} Sigma^t P x_0, or equivalently (P x_t) = Sigma^t (P x_0). This equation is dramatically simpler, since Sigma is diagonal, so if Sigma = diag(s_1, ..., s_n), then Sigma^t = diag(s_1^t, ..., s_n^t). In other words, this transformation "disentangles" the different components of A into ones that act independently. Here, the transformation x -> P x is what is called a "change of basis".These repeated matrix applications are common in physics, where they represent how a dynamical system evolves over time. The main difference is that in physics, the system evolves continuously, but similar transformations can be applied to solve these problems.
 This post really illustrates why HN needs MathJax support, and/or browsers need MathML support. Math notation is hard enough to read when you're reading the real version... when you're forced to read someone's ad-hoc "notation that I'm forced to use because I can't use real notation" it just becomes that much worse.Luckily, Chromium is getting MathML support (finally) unless something weird happens, which (presumably) means that Chrome and Edge will inherit that support as well. Firefox already has MathML, so hopefully we'll soon be in a position where three of the most widely used browsers support MathML.Still, it would be interesting to ask the Powers that Be if they'd be willing to implement MathJax here on HN....
 The problem with Linear Algebra specifically, is that it can be viewed from many different perspectives.The article here focuses on an "operational" perspective, how the numbers get added or multiplied together to turn into other numbers. However, Linear Algebra is also useful in geometry, and other situations.This "intuitive guide" to linear algebra sets you up very nicely for figuring out how to add and multiply matricies together. But it doesn't give you any intuition about a rotation (aka quaternions) in 3d space, for example. A lot of math books make the mistake of trying to teach all the perspectives at the same time, instead of focusing on just one viewpoint until the student gains mastery.A Quaternion is "just" a 4x4 matrix that represents rotation in 3-dimension space. Because you only move 3-ways rotationally (yaw, pitch, and roll), you're "underconstrained" with regards to the 4x4 matrix. Etc. etc. A lot of geometry intuition needs to be built here to really understand Quaternion... and none of that geometry is explored in the blogpost.Which is fine. Focus is good. But when people approach Linear Algebra, its important to know that its "so useful" that there are too many ways of looking at Linear Algebra... too many different, yet equivalent, understandings of the subject.
 I used to think this, my problem was that I wasn’t doing the exercises, instead just reading articles and watching videos and trying to get some kind of theoretical understanding. Everything makes a lot more sense once you’ve slogged through a bunch of repeated exercises.
 100% Agreed. I've written 3D engines in shipping games and yet I can't do math for shit. I've tried watching 3blue1brown and they have pretty pictures but they don't help at me all. I feel like they're mostly appreciated by people who already understand and can't remember what it's like to not understand.I was recently watching videos and trying to read papers on geometric algebra and getting totally and utterly lost on actually applying it.
 Higher math involves a very different way of thinking from the typical, useful things people do for a living. Exactness is important. The abstractions can run very deep. It's easy to get lost in the pure side of things without really understanding how to apply it.
 I think this is possibly the crux of it for me. I've certainly got well developed abstract thinking for software development, but software has always had a clear application for me, so the abstract thinking developed as a matter of course.I've rarely found any higher math instruction which takes for the form, "so you have this specific problem X, here is how we can solve it with technique Y"[1]. But I suspect that it is because it is higher math (which presumably means 'higher order' math).Without this, and without an inherent enjoyment of the pureness of the math, it seems somewhat esoteric for me personally. I'm not complaining, nor do I really think it should be any other way. I'm just reflecting on it really.This also makes me think of my foray into monads: "The thing about monads is once you finally understand them you immediately lose the ability to explain what they are to others." Not saying that's the case here, just feels related.[1] At least where I found problem X to be satisfactory. I didn't find my lecturer's problem of, "you're stood on a mountain described by this PDE, on what vector must one walk in order to stay at the same altitude" to be very applicable. I was a pretty wilful student though.
 Software development never gets that abstract. Yes there are abstractions, but compared to college math, they're extremely simple. The most complex abstractions I've come across in software engineering don't hold a candle to some of the abstractions you would see in a typical undergraduate math degree. Heck, my university used Baby Rudin [0] for its introductory analysis class, which was often taken by freshmen or sophomores.
 Personally I really struggle with the syntax and notation of upper-level math courses, I need a big cheat sheet of all the terms. It's like programming where we use i, j, k for loop variables instead of something more descriptive. My brain has to do one extra layer of translation between what's written and the concept being taught and then I lose focus, but intuitively I've always been fine with math concepts. When I took linear algebra in college (and did terrible) I particularly struggled with all the syntax that was introduced. Would love some tips if anyone else has a similar problem.
 That is to a large extent a matter of practice. The notation can become second nature and then really helpful - you can write down regression (ordinary least squares) in two lines, basically (including problem and solution), and it all makes sense and translates fairly directly into an algorithm.
 I'm currently working on a linear algebra heavy linear programming model for an optimization thesis and the general trick is, as with writing code, to improve iteration by iteration of the model. Starting with a complex mathematical model is always a bad idea, so start small and iteratively improve it. This means no use of advanced concepts unless needed. Also, nobody uses every bit of math in the everyday work. As with programming, you just need to understand the general concepts and the rest can be figured out step by step.
 Try Gilbert Strang's course at MIT, it's publicly available in video format, and starts from basic principles.
 I think the point where it gets confusing is where they stop showing you how to do calculations. With programming, you are always calculating something, even if you abstracted away from it.Geometric algebra seems more practical than most subjects but even its introductions suffer from this.
 If this interests you, you should check out the bivector community https://bivector.net/.Join the discord https://discord.gg/vGY6pPk.Check out a demo https://observablehq.com/@enkimute/animated-orbitsAlso at the end of February, there is geometric algebra event in Belgium. https://bivector.net/game2020.html All the big names in the field will be there.
 This seems to be "what are matrices and how do you work with them" and not linear algebra. I mean that can be useful sometimes but seems more like something you would teach in a numerics course instead.Actually, I think this way of explaining and motivating things (linear map==matrix) will get really, really confusing once you try to understand changes of bases or eigenvalue decomposition. A linear map is something that takes vectors and spits out vectors while preserving the vector structure (i.e. addition and scalar multiplication on the input give you addition and scalar multiplication of the output).
 A thread from 2015: https://news.ycombinator.com/item?id=8920638Discussed at the time: https://news.ycombinator.com/item?id=4633662
 Goes to show how different people have different tastes. I find this type of exposition very confusing and very unenlightening. Give me a Landau-style "minimalistic"/"focused" explanation any day. Not to mention, it tries so hard to simplify things to a simple analogy (the spreadsheet thing) that it ends up being plain misleading. In other words: "Make things as simple as then can be, but no simpler."
 Viewing Linear Algebra as the study of linear operators instead of matrices makes everything so much simpler.Of course AB != BAComposition makes senseInverse makes senseThis is the book that helped me get it http://linear.axler.net/
 I really like "matrices are the coordinate form of linear transforms." In the LA class I took the professor made a pretty big deal out of that, first by defining "lexicographic matrix basis" so he could write out matrices as vectors and then talking about mapping between the three different ideas.There's still stuff he said that I'm unpacking today... that was a dense class.
 This is a great book. I also recommend Halmos "Finite Dimensional Vector Spaces". The typical way linear algebra is introduced does not present a matrix as a linear transformation first and foremost.
 I'm reading Strang's linear algebra book and he teaches it in terms of combining columns and rows which I think is a lot clearer than explaining it in terms of linear equations.
 I love explainers like this, but it frankly makes me a little angry that the vast majority of the math teachers I had in highschool and college taught in the awful way described in the setup to the piece.Why is that? Has anyone studied it, or is there even a solid anecdotal explanation? The best one I can imagine is many of these professors simply don't care much for teaching and are more focused on their research, which is still infuriating but at least an explanation.I ended up with an undergrad in applied math, though I'm a software engineer now. I like math, but I feel like I never got to be all that great at it. I suspect I would've enjoyed it more and achieved more with explanations like these.
 How much of this explainer however seems better to us precisely because of the more comprehensive knowledge and understanding we already have?For example the author uses the word function liberally in the explanation. However, when studying functions in school in math it was super complicated for me. It was only after I started programming, learning the programming language meaning of function, and then when I was reintroduced to mathematical functions through functional programming that I truly grasped mathematical functions and all the stuff I was taught in school.I’m not arguing that school level math does not need to be improved. I just think we should be cautious that because an explanation that seems intuitive to us now after having gained a complete introduction to all math concepts as well as programming (esp on HN) may not necessarily be as intuitive when students who are only exposed to a very small subset of mathematical concepts encounter them.
 For sure there is an effect on Reddit and other places where someone will post a question such as, "I'm having trouble with my Calc class, what is a good book?" and people seriously answer Calculus on Manifolds.Now, CoM is a classic, a real great book, but it is useful only to people who have reached a certain level of mathematical maturity. That, presumably, is not the questioner.A version of this is that I also see people who write, "I didn't understand this topic when I took the course but now years later, I see it is all actually very easy."(I call this Second Book Syndrome because I don't know of a common name for it. I understand this to be what Zen people mean by the "Gateless Gate," that after struggling with something at great length a person can come to see that there is no real difficulty. But I've never heard anyone else apply that name to this phenomenon and I'm not Zen trained so I'm not sure that is right either.)
 Once you understand monads, you lose the ability to explain monads. Hence the number of monads tutorials grows at an exponential rate as every new understander tries to explain them and fails.it's a fun problem in teaching
 > Once you understand... you lose the ability to explainSorry, this does not make sense to me.
 Douglas Crockford on Monads:In addition to its being good and useful, it’s also cursed and the curse of the monad is that once you get the epiphany, once you understand - "oh that's what it is" - you lose the ability to explain it to anybody else.
 Here is a link that helps me think about it: https://www.alanwatts.org/3-3-10-gateless-gate/ (I teach Math and a common complaint, against all math teachers, is "They obviously know the material but they cannot teach it." When I first heard AW talk about this it was a relevation for me because it so closely linked with my experience. You say to people "A vector space is a place for linear combinations to happen" and they don't get it, of course they don't get it, I wouldn't have gotten it, but that's what a vector space is. So you have to do lots of examples, and work around the edges, and somehow sit with it for a while. Anyway, I find that for me this link conveys the point of that quote about monads.)
 Ironically, I think the original paper that introduced Monads as a useful computational abstraction is the clearest explanation I've seen.
 Corollary: a good teacher hasn’t understood anything. Well, that’s wrong, I think.However, this might be true: A good teacher remembers not having understood it.
 Actually a rather famous example of your point are the Feynman lectures. They are often hailed as a great, "easy" and intuitive explanation of physics. However, supposedly at the time, the undergrad students Feyman was teaching did not think their were all the rage. It was actually the grad students who retook the material and really enjoyed them. To support your point I think sometimes it is really necessary to have been exposed to the material and been able to grasp some part of it, then you are open to these type of explanations which really lift you over the hurdle of the things that you didn't quite get before.
 As someone who learned to program before taking any interesting math I was always so confused by the amount of early math classes spent explaining function notation.It wasn't until I tried teaching people that I realized how odd the notation can seem.
 you should be happy someone explained it to you- I feel that notation was overlooked tremendously in my math education- I actually just brought this up in a thread yesterday. It was like "oh, we have this dx now... k."
 I agree with your post. I had the same problem where match was harder for me to comprehend. It wasn't until I started programming that I started to understand the concept you cite (Functions.).
 Have you ever taught?I taught Astronomy to the people in the years below me (a school tradition because it was optional), and it was absolutely exhausting trying to plan good lessons that didn't involve getting them to memorise stuff by wrote. I ""derived"" Kepler's laws for a bunch of 14-15 year olds who didn't even know logarithms yet and it was pretty brutal intellectually.Also, I think teaching mathematics "intuitively" requires a bit of cooperation from the student - not in the sense of intelligence, just that for for every guy (or girl) who watches 3Blue1Brown (and looks deeper into the pure mathematics) there's another who's just along for the ride. I think the frequencies of those personalities are a product of how they were taught, but it's very difficult to convince people in "teaching time" as opposed to naturally (I was in the lowest maths group for years until I picked up a calculus book on a whim, but no teacher could've convinced old me that mathematics can be beautiful)Quick Edit: To quote Tim Minchin, "Be a teacher". I might come across as moaning in the above but it's really interesting to explain things (in my experience at least - I love writing documentation!), and a good bullshit-test on yourself
 I had an eye opening experience and some closure about my own eduction after reading a certain book, a collection of short biographic accounts of famous Germans and their horrific experience in school during the last few centuries. School has always been bad and actually used to be borderline insane, and yet these great characters, politicians, artists and scientists still developed fine.Rant: If society gave a shit teacher would be the highest paid job with the highest standards, both in knowledge and in teaching skill. I mean literally 500k/year for the most noble calling on earth. The kind of progress humanity would make with a generation taught by the best, and teaching itself revolutionized, would be crazy.
 Here are two reasons I think are important, there are more but these stand out. (1) Teaching is hard, and the amount of training you get in teaching higher level maths (rather than the just learning higher level maths itself) is very limited. (2) intuition in maths will lead you on a merry path to very wrong ideas. There are functions that are continuous everywhere without a derivative anywhere, Russell's paradox, classically zenos paradox, etc, etc. A big part of higher level maths training is in being precise with what you mean so that you can learn to do proofs and not trick yourself. I think a synthesis of these approaches is the sweet spot. Be precise but show applications to motivate the material.
 I think the problem might be that the kids best at math become teachers. The ones that it ”came to easily”.This creates the not-so-obvious problem that the easier you learn something the harder it might be to teach it to someone who doesn’t get it.If you on the other hand struggle hard to find a way to understand a complex concept, you might be good at transfering that knowledge forward.
 Already this is so helpful, as someone with only a partial high school math background. The visual "pouring" analogy alone made it worth the read.
 I remember taking Advanced Algebra Honors in 10th grade. It was basically Algebra II with a few (seemingly teacher-selected based on the experience of students who had a different teacher) advanced topics thrown in. One of them was matrices, and I was completely stumped by them. I now encounter them all the time, and wish I'd been able to wrap my head around them when I was younger.
 20+ years ago I combined learning JavaScript and Linear Algebra https://mkaz.blog/math/javascript-linear-algebra-calculator/
 Thank you for writing this up, or reposting.I agree that it is better to understand math, and computer science, intuitively first. Learning the basics instead of learning how to think in them forces memorization and is frankly in a time gone by.If only I could've been taught this way when I was younger, then I'd actually be any good at any advanced math.
 The only way to test if you've 'understood' something is to apply it to real-world problems.If it works - then you understood it. If it doesn't - then you didn't."Understanding" without way of external verification seems no different to dopamine-chasing.Measuring on output and all that...
 The only way to test is if you talk with an expert and he says you have understood. There are many things in linear algebra that you can use in practice even when you didn't really understand them. This is the reason why self-studying certain topics is very hard, you still need (good) teachers to give you constant feedback.
 That seems like a verbalistic notion of 'understanding'.How do I falsify the expert's claims about my understanding?
 Speaking for Linear Algebra, I learnt more reading for a few hours the appendix of "The Design of Rijndael: AES - The Advanced Encryption Standard" than I did in 6 months of theoretical university teaching full of useless technical terms and solutions in search of problems...
 > solutions in search of problems...This sounds like it was meant to be pejorative, but it's what (applied) linear algebra, and applied mathematics more generally, is. Anyone can learn about a certain mathematical topic upon realising it's the one relevant to the problem they're facing—and learn it way more quickly, due to motivation and focus, than they would in a general-purpose course on the topic; the art is in recognising what mathematics is relevant, and you can't do that if you've never heard of it before. Having a library of (conceptual, not cookbook) solution hooks on which to hang your problems is how you get to be good at using mathematics.
 Of course it was meant to be pejorative and that pretty much summarize my experience through university. In this particular case, the teacher wasn't giving a rat arse about his class and was merely there for the safe job, pension, sprinkles by some "research".Anyhow, I disagree with that top-down approach, which seems to be very... European. I much prefer to follow a more logical path where the problem preclude the introduction to the solution.
 > Anyhow, I disagree with that top-down approach, which seems to be very... European. I much prefer to follow a more logical path where the problem preclude the introduction to the solution.I agree that it's easier to learn that way, but it's not much use when solving real-world problems; you can learn the techniques relevant to the solution once you know what they are, but, if you haven't met them before, you won't recognise that they're the right ones.
 OP Here:I just wanted to give some context to how I found this page, and why I thought it would be good to post.I may be putting myself on the spot here: I never took a linear algebra course in undergrad. It was a heavily encouraged option, of course, but I felt I understood the basic rules enough to not really need formal study. I opted to study other areas, partially motivated by a fear I wouldn't do well and would hurt my GPA (my god was I vain, I feel I could do so much more studying full-time with my current world-view).As time has gone on, and ML and quantum computing have simply blown up since I graduated in 2014, I quickly realized the magnitude of my mistake. I have frantically self-studied for years to try to make up the gaps in my mathematical understanding, and linear algebra has come up time and time again. I can do the processes, but they never clicked, I had no intuition.I want to help others in my position cut to the chase, and study the highest yield, most intuition giving resources.I actually developed the mental model shared in this guide on my own, and was positively delighted to find to this while thinking over a comment I was drafting on here. This page lays things out so clearly. The component steps are intuitive and I can commit them to memory/recall what they mean without needing to dig up my notes to self!===I find this page gives an excellent foundation, and goes great with these resources:* A web site which clearly shows how to do matrix multiplication in a way that's easy to recall, it makes the procedure like riding a bike:http://matrixmultiplication.xyz/Huge thanks to Jeremy Howard of fast.ai for mentioning this in one of his lectures, this tool is how I finally got matrix multiplication to click* A paper named "An Introduction to Quantum Computing" (bear with me, it's superbly well written and very approachable):https://arxiv.org/abs/0708.0261Page 3 of that paper lays out matrix multiplication (e.g.: applying a "transformation matrix" in the spatial parlance of 3blue1brown's videos) as a traversal of a directed graph. A very useful understanding, and shows how generalizable the tools of linear algebra really are in my opinion.* The essence of linear algebra, by 3blue1brown (fantastic for a geometric/"transformation of space" view of linear alg):
 This is a nice resource - I wrote one myself as well which is mostly based on the series by 3Blue1Brown, as well as other resources which I found useful and which used a visual approach to introducing linear algebra.You can find my guide here:
 http://immersivemath.com/ila/index.htmlAn interactive Linear Algebra course to complement the article.
 I love betterexplained.Does this guy have a patreon?
 He has a business that sells courses and books right on the homepage.
 Are there resources like this for Calculus/Prob/Stats?
 Does something like this exist for differential equations?
 May be this can help you:https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVl...This is 3Blue1Brown's video series on Differential Equations. Do support him on Patreon if this really helps you.

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27 May 2011, 18:48
What do you want to do? It really can't be based that simple. Haas is a really well respected school. I am sure Owen is fine, but depending on where you want to get it might make a big difference.
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### Week of November 28, 2021

 12:00pm - Zoom ID 989 6529 0738. Passcode: the last four digits of the zoom ID in the reverse order - Probability and Analysis Webinar Joe Neeman - (UT Austin) Large deviations for triangle densities Take a uniformly random graph with a fixed edge density e. Its triangle density will typically be about e^3, and we are interested in the large deviations behavior: what's the probability that the triangle density is about e^3 - delta? The general theory for this sort of problem was studied by Chatterjee-Varadhan and Dembo-Lubetzky, who showed that the solution can be written in terms of an optimization over certain integral kernels. This optimization is difficult to solve explicitly, but Kenyon, Radin, Ren and Sadun used numerics to come up with a fascinating and intricate set of conjectures regarding both the probabilities and the structures of the conditioned random graphs. We prove these conjectures in a small region of the parameter space.   Joint work with Charles Radin and Lorenzo Sadun 2:00pm to 3:00pm - 510R - Combinatorics and Probability Olya Mandelshtam - (University of Waterloo) The multispecies zero range process and modified Macdonald polynomials Over the last couple of decades, the theory of interacting particle systems has found some unexpected connections to orthogonal polynomials, symmetric functions, and various combinatorial structures. The asymmetric simple exclusion process (ASEP) has played a central role in this connection. Recently, Cantini, de Gier, and Wheeler found that the partition function of the multispecies ASEP on a circle is a specialization of a Macdonald polynomial $P_{\lambda}(X;q,t)$. Macdonald polynomials are a family of symmetric functions that are ubiquitous in algebraic combinatorics and specialize to or generalize many other important special functions. Around the same time, Martin gave a recursive formulation expressing the stationary probabilities of the ASEP on a circle as sums over combinatorial objects known as multiline queues, which are a type of queueing system. Shortly after, with Corteel and Williams we generalized Martin's result to give a new formula for $P_{\lambda}$ via multiline queues.   The modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ are a version of $P_{\lambda}$ with positive integer coefficients. A natural question was whether there exists a related statistical mechanics model for which some specialization of $\widetilde{H}_{\lambda}$ is equal to its partition function. With Ayyer and Martin, we answer this question in the affirmative with the multispecies totally asymmetric zero-range process (TAZRP), which is a specialization of a more general class of zero range particle processes. We introduce a new combinatorial object in the flavor of the multiline queues, which on one hand, expresses stationary probabilities of the mTAZRP, and on the other hand, gives a new formula for $\widetilde{H}_{\lambda}$. We define an enhanced Markov chain on these objects that lumps to the multispecies TAZRP, and then use this to prove several results about particle densities and correlations in the TAZRP. 4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics Eun-Jae Park - (Yonsei University) Staggered discontinuous Galerkin methods on polygonal meshes Recently, polygonal finite element methods have received considerable attention. This is because general meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and  non-matching interfaces.  In this talk,  a new computational paradigm for discretizing PDEs  is presented via the staggered Galerkin method on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes. We derive a simple residual-type error estimator. Numerical results indicate that optimal convergence can be achieved for both the potential  and vector variables, and the singularity can be well-captured by the proposed error estimator. Then, some applications to diffusion equations, Stokes equations, and linear elasticity equations are considered. Finally, we extend this approach to high-order polynomial approximations on general meshes.
 4:00pm - NS2 1201 - Differential Geometry Hongyi Sheng - (UC Irvine) Deformations of the scalar curvature and the mean curvature In Riemannian manifold $(M^n, g)$, it is well-known that its minimizing hypersurface is smooth when $n\leq 7$, and singular when $n\geq 8$. This is one of the major difficulties in generalizing many interesting results to higher dimensions, including the Riemannian Penrose inequality. In particular, in dimension 8, the minimizing hypersurface has isolated singularities, and Nathan Smale constructed a local perturbation process to smooth out the singularities. However, Smale’s perturbation will also produce a small region with possibly negative scalar curvature. In order to apply this perturbation in general relativity, we constructed a local deformation prescribing the scalar curvature and the mean curvature simultaneously. In this talk, we will discuss how the weighted function spaces help us localize the deformation in complete manifolds with boundary, assuming certain generic conditions. We will also discuss some applications of this result in general relativity.
 2:00pm to 3:00pm - Rowland Hall 510R - Combinatorics and Probability Roman Vershynin - (UCI) Mathematics of synthetic data and privacy An emerging way to protect privacy is to replace true data by synthetic data. Medical records of artificial patients, for example, could retain meaningful statistical information while preserving privacy of the true patients. But what is synthetic data, and what is privacy? How do we define these concepts mathematically? Is it possible to make synthetic data that is both useful and private? I will tie these questions to a simple-looking problem in probability theory: how much information about a random vector X is lost when we take conditional expectation of X with respect to some sigma-algebra? This talk is based on a series of papers joint with March Boedihardjo and Thomas Strohmer, mainly this one: https://arxiv.org/abs/2107.05824
 9:00am to 10:00am - Zoom - Inverse Problems Yi-Hsuan Lin - (National Yang Ming Chiao Tung University) TBA https://sites.uci.edu/inverse/ 10:00am to 10:50am - Zoom - Number Theory Ofir Gorodetsky - (Oxford University) TBA 11:00am to 11:50am - Zoom ID: 949 5980 546, Password: the last four digits of ID in the reverse order - Harmonic Analysis Gabriel Rivière - (Université de Nantes) Meromorphic continuation of Poincaré series Poincaré series are natural functions that arise in Riemannian geometry  when one wants to count the number of geodesic arcs of length less than  T between two given points on a compact manifold. I will begin with an  introduction on this topic. Then I will discuss some recent results with  N.V. Dang (Univ. Paris Sorbonne) showing that, in the case of negatively  curved manifolds, these series have a meromorphic continuation to the  whole complex plane. This can be shown by relating Poincaré series with  the resolvent of the geodesic vector field and by exploiting recent  results on this resolvent obtained through microlocal methods. If time  permits, I will also explain how the genus of a surface can be recovered  from the analysis of these series. 4:00pm - RH 306 - Colloquium Jinchao Xu - (Penn State University) TBA


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Elementary Technical Mathematics

$0.115150$
To multiply the two numbers, the decimals can be initially ignored to obtain $4606 \times 25$. Multiply the numbers to obtain: \begin{array}{ccc} &&\space \text{ }4606 \\&\times &\underline{\space \text{ } \space \text{ }\space \text{ } \space25} \\&&23030 \\&&\underline{\space 9212\space \text{ } \space} \\&&115150\end{array} The first factor has 3 decimals places and the second factor has $3$ decimal places. Thus, the product has $3+3=6$ decimal places. Therefore, the product is $0.115150$.


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# The spectral evolution of nebular phase from Nova V5668 Sgr

### Abstrak

The spectral evolution of the Nova V5668 Sgr has been observed on 12 June, 23 June, and 15 August 2015 using NEO-R1000 spectrograph attached to the C-11 telescope at Bosscha Observatory, Lembang, Indonesia. The observed spectra indicate that the nova has been entering the nebular phase. The spectra during this phase showed strong forbidden lines of [O I] $\lambda$6300, 6364, [O II] $\lambda$7320, and [O III] $\lambda$5007, [N II] $\lambda$5755. The spectra also showed emission line of Fe II $\lambda$4924, 5169, 5317, 6149. We classify Nova V5668 Sgr into the Fe II type. We present and discuss the variation of nebular abundance N(X)/N(H) of the Nova V5668 Sgr in the light of its evolutionary stages.

Publikasi
Journal of Physics: Conference Series, Volume 771, Issue 1, article id. 012054 (2016)


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# Constructing likelihood ratio test

Let $y_1$ and $y_2$ be independent Poissons with means $\lambda_1$ and $\lambda_2$. Find the likelihood ratio statistic for the test $H_0: \lambda_1 = \lambda_2$. Specify the asymptotic null distribution, and describe the condition under which the asymptotics apply.

I have the answer. It's $2\sum_i y_i\log\left(\frac{y_i}{ \bar{y}}\right)$. The asymptotic distribution is chi-squared with $df = 1$, when $\lambda_1$ and $\lambda_2$ are large.

The problem is that I have trouble arriving at this answer. I start with the poisson pmf, find the likelihood for each and take the ratio. But I dont see where the $\bar{y}$ comes from? Can someone show me how to get this? I'd really appreciate the help.

Edit: OK, what I got after taking the ratio of the likelihoods (at mle's) was $2n [\lambda_2 - \bar{y} + \bar{y}log(\bar{y} / \lambda_2)]$

• 1. Unless I made an error, I think the answer you give appears to be wrong; I think it's missing a term that can't be dropped if you're using the asymptotic approximation. 2. Note that $y_1+y_2=2\bar{y}$. – Glen_b Nov 15 '16 at 8:49
• Please add the self-study tag, read its tag-wiki and modify your question to follow the guidelines on asking such questions. In particular, you'll need to clearly identify what you've done to solve the problem yourself, and indicate the specific help you need at the point you struck difficulty. – Glen_b Nov 15 '16 at 8:49
• To add the tag you'd need to remove one. I think "generalized-linear-model" or "maximum-likelihood" could go, but if you want to keep both, just leave it out. – Glen_b Nov 15 '16 at 9:46
• Thanks Glen- see my edits. Also, the answer I gave came from my book so I'm not sure...but I haven't matched it either. – kiring24 Nov 15 '16 at 11:33
• What happened to the factorial terms? – Glen_b Nov 15 '16 at 11:39

Leaving a little work at the end for you, I got the following. This is a bit loose, but it's intended as an outline -- I'm sure you can tidy it up. Anyway here goes:

$\Lambda = \frac{\mathcal{L}_0(y_1,y_2)}{\mathcal{L}_1(y_1,y_2)}$

$\mathcal{L}_1=\exp(-\lambda_1)\lambda_1^{y_1}/y_1!\cdot \exp(-\lambda_2)\lambda_2^{y_2}/y_2!$

$\mathcal{L}_0=exp(-2\lambda)\lambda^{y_1+y_2}/(y_1+y_2)!$

$\hat{\lambda}_1=y_1\,,\hat{\lambda}_2=y_2\,,\hat{\lambda}=(y_1+y_2)/2=\bar{y}$

$\log(\Lambda)=\ell_0(y_1,y_2)-\ell_1(y_1,y_2)$

$= -2\lambda +2\bar{y}\log(\lambda)-\log((y_1+y_2)!)\\ \qquad - [-\lambda_1+y_1\log(\lambda_1)-log(y_1!)-\lambda_2+y_2\log(\lambda_2)-log(y_2!)]$

at MLE

$\log(\Lambda)=-2\bar{y} +2\bar{y}\log(\bar{y})-\log((y_1+y_2)!) \\ \qquad - [-y_1+y_1\log(y_1)-log(y_1!)-y_2+y_2\log(y_2)-log(y_2!)]$

$=-2\bar{y} +2\bar{y}\log(\bar{y})-\log((y_1+y_2)!)\\ \qquad - [-y_1+y_1\log(y_1)-log(y_1!)-y_2+y_2\log(y_2)-log(y_2!)]$

etc

If you simplify further, that gets to the answer in the question plus an additional term I don't believe you can leave out.


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If 5x-2y = 17 and 3x +y =8, th...

If 5x-2y = 17 and 3x +y =8, then find the values of x and y.

Updated On: 27-06-2022

Text Solution

Answer : x = 3 and y=-1.


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# Ladies And Gentlemen Part I (Album Version)

Ladies And Gentlemen Part I (Album Version)

## Track by The Presidents Of The United States Of America

62
0

LYRICS

Good evening ladies and gentlemen, hope you enjoyed the show Good bye ladies and gentlemen, it's time to go Good evening ladies and gentlemen, that's all the songs we know Good bye ladies and gentlemen, so let go! let's go! Let go! let's go! Let go! let's go! Let go! let's go! Let go! let's go! Let go! let's go! Let's go! Let's go! Let's go!

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The structure of a procurement group will generally depend on several key factors around an enterprise that is going to order procurement items for a project. That is why organizational structures are a first-class citizen in our framework. A typical abstract example of such structures is the so-called “vertical differentiation” or “authority structure” of organizations, usually considered to be a “hierarchy” structure. = [\! (2004). If an agent fails in performing one of the stated or delegated tasks, a kind of supervisor agent should engage in the performance of that task: organization calls for a form of supervision activity (Giddens 1984). We choose however not to complicate matters further and we assume in this work that all agents have the capabilities needed to perform the tasks they are appointed to. An agent does something causally blameworthy, if it is causally responsible and if it knows that the action it performs leads to a violation which could be avoided by not performing the action. Control Responsibility issues arise any time a group of agents acts collectively in order to achieve certain objectives. [ X: \alpha_{2} ]\!] These changes mean that HR departments play a more strategic role within an organization, and that manager and director-level HR roles are interacting with the CEO, CFO or COO of the company. Any changes made will need to take into account resource constraints and the size, nature and needs of the business and customers. [ X: \overline{\alpha} ]\!] Delegation of authority must be commensurate with the responsibility assigned. In other words what delegate actions do, with respect to all the other actions being performed in the step, is just creating obligations, given that the necessary preconditions hold. (Semantic constraints for Formula 23 is proven since there always exists a model satisfying $$O(Plan(Ag,\tau)) \wedge \neg [a_1: \alpha_1 ; \ldots ; a_{i}: \alpha_{i}] {\mathfrak{TaskR}}(a_i,\alpha_i)$$, which intuitively corresponds to the model in which the task-based responsibility ceases to hold after the execution of the relevant event. Since agents’ tasks can be accomplished or not, the monitoring activity is an indispensable activity in which any organization has to engage. Computer science technical report 04-45, University of Massachusetts, May 2004, Jones AJI, Sergot M (1996) A formal characterization of institutionalised power. i https://www.project-management-skills.com/organizational-structure-types.html Negation of sequences constitutes a delicate matter. 4.1). A subset of Evt is of particular interest for our purposes, that is the set of events of the form $$X\;:\;\alpha$$ such that X is a singleton and α is an organizational action in $${\mathcal{A}}$$. In its simplest form, control consists in a monitoring activity triggering appropriate reactions to determinate failures or violations. If an agent is found to be causally responsible for a violation by not performing β, then if an appropriate control structure is in place and a monitor action is performed to check whether β has been performed then the monitoring agent becomes task-based responsible for that action. ]\doublecup [\! Organizational Structure and roles 1. In other words they guarantee the group ‘to act in an organized way’. Organizational Development field is required to implement and oversee methodologies to strengthen employee abilities and knowledge, increase efficiency, and improve leadership to maintain the overall health of an organization. Springer, Berlin, pp 30–40, Dunin-Keplicz B, Verbrugge R (2002) Evolution of collective commitment during reconfiguration. The key idea consists in enabling the possibility to formalize not only notions concerning the activity of groups of agents, but also notions concerning the activity of organized groups of agents, that is to say of groups of agents displaying some form of organizational structure. Copyright 10. with the agent indexes a Responsibilities, instead, concern agents and arise in relation with task-allocation and structure once there are agents enacting the roles of a given organization. 2 the notion of organizational structure is extensively discussed and the analytical perspective presented in Grossi et al. Role of Organization Structure: An organization structure serves various functions of the business. Organizational structure is the chain of hierarchy, which divides entire employees of an organization based on their level, roles, and responsibilities. In this case the relevant structural question is: who can (successfully) enable or empower whom? The activities are placed according to their importance in the structure and proper guidelines are given for resource allocation. Notice finally that event expressions consist of an action expression (α) plus an index denoting a group of agents (X). Springer Verlag, Berlin, pp 61–80, Dignum F, Meyer J-J Ch, Wieringa R, Kuiper R (1996) A modal approach to intentions commitments and obligations: intention plus commitment yields obligation. [ \xi_{1} ; \xi_{2} ]\!] There are a … The authority is delegated to the lower levels of various departments and responsibility is fixed. are three irreflexive binary relations on Roles characterizing the Power, Coordination and Control structures. This motivates the following semantic constraint. For strategy to be successful, an organization will need to clearly define the roles and r… Revised Selected Papers, volume 3228 of Lecture notes in computer science. You will be provided with simple clear documents that can be issued as part of a health and safety induction or to re-induct existing employees and the document can be used to set key performance targets. Span of control refers to the number of people a manager can directly supervise. , R A well designed structure will help both management and operation of a business. We can now introduce the operations that constitute the semantic counterpart of our syntactic operators. The possibility of delegating goals constitutes one of the essential aspects of organizations. In organizations an agent can happen to be causally responsible of some failure without actually being accountable for that in the eyes of the organization. Work on organizations (especially in MAS The agent would be considered causally responsible, but it would not be considered blameworthy. The transfer takes place in the form of a directed obligation (Dignum 1999) of the agent enacting the first role to the agent enacting the subordinated one. :α In other words a role-based plan $$Plan(AR, \tau)$$ and a role enactment configuration Rea univocally determine an agent-based plan, i.e., a complex event description. If none of the preconditions hold, the monitor action does not influence the transition generated by the step. The following are validities of the framework: Proofs are given showing that countermodels are impossible.

## organizational structure roles and responsibilities

Ieee Vtc Acceptance Rate, Cafe And Me Menu, Added Value Marketing, Frigidaire Door Closing Cam, Marantz Pm5005 For Sale, Peg Perego Follow Me Ice,


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# Policy Gradient not "learning"

I'm attempting to implement the policy gradient taken from the "Hands-On Machine Learning" book by Geron, which can be found here. The notebook uses Tensorflow and I'm attempting to do it with PyTorch.

My models look as follows:

model = nn.Sequential(
nn.Linear(4, 128),
nn.ELU(),
nn.Linear(128, 2),
)

Criterion and optimisers:

criterion = nn.BCEWithLogitsLoss()
optim = torch.optim.Adam(model.parameters(), lr=0.01)

Training:

env = gym.make("CartPole-v0")

n_games_per_update = 10
n_max_steps = 1000
n_iterations = 250
save_iterations = 10
discount_rate = 0.95

for iteration in range(n_iterations): # Run the game 250 times
all_rewards = []
n_steps = []
for game in range(n_games_per_update): # Run the game 10 times to accumulate gradients
current_rewards = []
obs = env.reset()
for step in range(n_max_steps): # Run a single game a maximum of 1000 steps

logit = model(torch.tensor(obs, dtype=torch.float))
output = F.softmax(logit, dim=0)
c = Categorical(output)
action = c.sample()

y = torch.tensor([1.0 - action, action], dtype=torch.float)
loss = criterion(logit, y)
loss.backward()

obs, reward, done, info = env.step(int(action))
current_rewards.append(reward)
if done:
break
n_steps.append(step)

all_rewards.append(current_rewards)

# Performs the discount and normalises
all_rewards = discount_and_normalize_rewards(all_rewards, discount_rate=discount_rate)

# For each batch of 10 games multiply the discounted rewards against the gradients of the
# network. Then take the mean for each layer
means = []
for game_index, rewards in enumerate(all_rewards):
for step, reward in enumerate(rewards):

# Apply the new gradients to the network
for p, g in zip(model.parameters(), new_gradients):
optim.step()

When I run the code for 250 interactions I print the average game length I get:

Iteration: 50, Average Length: 18.2
Iteration: 100, Average Length: 23.4
Iteration: 150, Average Length: 29.9
Iteration: 200, Average Length: 11.2
Iteration: 250, Average Length: 38.6

The network isn't really improving and training for longer doesn't help. My two questions are: 1. Is there anything obviously wrong that I'm doing? 2. I've notices the log of the probability is used in the tensorflow implementation, but I'm not sure how to integrate it here

I can't say for sure but I think the issue here is you're not subtracting the mean of the rewards.

The idea is that actions with above average reward are positive after mean normalization, while actions with below average reward are negative after mean normalization.

Your update step is -log(P(action))*reward, which you then minimize with your optimizer.

P(action)<1 therefore log(P(action))<0, -log(P(action))>0

If reward>0, -log(P(action))*reward>0. Minimizing this value is the same maximizing log(P(action))*reward<0, which is maximized when P(action)=1.

Conversely, if reward<0, -log(P(action))*reward<0. This has the opposite effect, where P(action) is driven to 0.

The important part is the different sign on above average/below average rewards causes actions associated with good rewards to have their probability increased, while actions associated with bad rewards have their probability decreased.


text:
Level 3 PSLE
Linda saved $240 in February. This amount was a 20% increase from what she saved in January. The amount she saved in March was a 30% decrease from what she saved in February. 1. What was the total amount that Linda saved from January to March? 2. What was the percentage increase in the amount Linda saved in March compared to January? 5 m Level 1 PSLE The number of visitors to a theme park was 15000 in July. This was a 20% increase from the number in June. How many visitors were there in June? 2 m Level 2 PSLE A shop sells shirts in four different sizes. The table shows the number of T-shirts sold in June for size S, M and L but not XL. 1. 50% of the shirts sold were in size M. How many of the shirts sold were in size XL? 2. In July the number of shirts sold in size XL remained the same but there was an increase in the number of shirts sold in the other sizes. Did the percentage of the shirts sold in size XL (1) increase, (2) decrease or (3) remain the same from June to July? Give the answer in number. (Eg 1) 2 m Level 2 PSLE Gerry was given a fixed amount of allowance each month. In January, he spent$80 and saved the rest. In February, he spent 10% less and his savings increased by 25%. How much was Gerry's allowance for each month?
2 m
Level 2
Fill in the blanks.
Bryan gives him more stickers.
The number of stickers that Adam has increases by 25%.

1. Number of stickers that Adam has at first = _____ u
2. Number of stickers that Bryan gives him = _____ u
3. Number of stickers that Adam has in the end = _____ u
3 m
Level 3
Fill in the blanks.
Adam and Bryan share some stickers.
When Adam gives 10 to Bryan, Bryan's share increases by 20%.

1. Number of stickers that Bryan increases by in percentage = _____%
2. Number of stickers that Bryan has in the end in percentage = _____ %
3. _____% of Bryan's stickers is the same as 10.
4. Number of stickers that Bryan has at first = _____
4 m
Level 3
Fill in the blanks.
Adam and Bryan share some stickers.

1. Number of stickers that Adam decreases by in percentage = _____%
2. Number of stickers that Adam has in the end in percentage = _____ %
3. _____% of Adam's stickers is the same as 10.
4. Number of stickers that Adam has at first = _____
4 m
Level 3
Fill in the blanks in whole numbers or decimals.
2 boys, Adam and Bryan, share some stickers.
Bryan has 20% of the total.
his number of stickers increases by 50%.

1. Number of stickers that Bryan has at first = _____ u
2. Change in the number of stickers that Bryan has = _____ u
3. Number of stickers that Bryan has in the end = _____ u
4. Number of stickers that Adam has in the end = _____ u
4 m
Level 3
Fill in the blanks.
The number of stickers decreases by 25%.

1. Number of stickers that Adam has at first = _____ u
2. Number of stickers that Adam gives away = _____ u
3. Number of stickers that Adam has in the end = _____ u
3 m
Level 2
Fill in the blanks.


text:
Let the centre of the circle be $O$ and let $A$ and $B$ be corners of one of the shaded squares, as shown.
As the circle has area $\pi$ square units, its radius is $1$ unit. So $OB$ is $1$ unit long.
Let the length of the side of each of the shaded squares be $x$ units.
By Pythagoras's Theorem, $OB^2 = OA^2 + AB^2$, that is $1^2 = (2x)^2 + x^2$.
So $5x^2=1$. Now the total shaded area is $8x^2 = 8 \times \frac{1}{5} = 1 \frac{3}{5}$ square units.


text:
Questions

# Network Drive Path ? Help

Locked

### Network Drive Path ? Help

Hi?

I have in a standalone Pc the following statements (working perfectly):
Dim wSht As Worksheet
Dim wBk As Workbook
Sheets("OnHand").Cells(5, 6) = 1
Set wBk = Workbooks("_14KbWC126Ot.xls")
Set wSht = Sheets("SchedulingPanel")
wSht.Copy before:=wBk.Sheets("OT")
Sheets("SchedulingPanel").Select
ActiveWindow.SelectedSheets.Delete
Sheets("SchedulingPanel (2)").Select
Sheets("SchedulingPanel (2)").Name = "SchedulingPanel"
Windows("_14KbWC126Sp.xls").Activate
Sheets("OnHand").Cells(5, 6) = 0

Wen I install the two Excel files in a server (Network Drive), I get the error code 9 "Subscript out of range" (because the Path is not correct) in the following statement:
Set wBk = Workbooks("_14KbWC126Ot.xls")

Can somebody help me and indicate to me how to define the Path in a Network Drive?

Thanks

## Where are you opening the files from?

Collapse -

I believe the issue may be related to where you are opening the files from.

I assume from your post that you copied the two files to a network share, and then attempted to launch them from your local machine. If that is the case, then you may want to provide the full network path in the statement that opens the workbook:

Replace this:
Set wBk = Workbooks("_14KbWC126Ot.xls")

With this:
Set wBk = Workbooks("\\YOURSERVER\YOURFOLDERPATH\_14KbWC126Ot.xls")

Or, if you have a mapped network drive, then do this:
Set wBk = Workbooks("X:\YOURFOLDERPATH\_14KbWC126Ot.xls")

Where X is your mapped network folder.

The issue is that although the file is sitting in the network folder, you are using your local copy of Excel to open it, so when Excel executes the VBA, it is looking for your file (_14KbWC126Ot.xls) in the executable directory (e.g. c:\program files\Microsoft Office\).

I hope that helps.

Regards,

KB


text:
## Rodrigues’ Rotation Formula

We can compute a rotation matrix $$R \in SO(3)$$ from an angle $$\theta$$ and axis $$l$$ (unit vector)

$R = I + \sin(\theta)C + (1-\cos \theta)C^2$

where $$C$$ is the antisymmetric matrix:

$C = \begin{bmatrix} 0 & -l_z & l_y \\ l_z & 0 & -l_x \\ -l_y & l_x & 0 \end{bmatrix}$

Our proof will use the Taylor's formula to define the exponential of a matrix $$M$$

$e^M = \sum_{k=0}^{\infty}\frac{M^k}{k!}$

where $$M^0$$ is defined to be the identity matrix $$I$$. Because $$C^3 = -C$$ and the Taylor expansions:

\begin{align}
\sin \theta &= \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots + \frac{(-1)^n}{(2n+1)!}\theta^{2n+1} + \cdots \\
\cos \theta &= 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots + \frac{(-1)^n}{(2n)!}\theta^{2n} + \cdots
\end{align}

then we obtain

\begin{align}
R &= e^{\theta C}\\
&= \sum_{k=0}^{\infty}\frac{(\theta C)^k}{k!} \\
&= I + \frac{1}{1!}\theta C + \frac{1}{2!}(\theta C)^2 + \frac{1}{3!}(\theta C)^3 + \frac{1}{4!}(\theta C)^4 + \frac{1}{5!}(\theta C)^5 + \frac{1}{6!}(\theta C)^6 + \cdots \\
&= I + (\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots)C + (\frac{\theta^2}{2!} - \frac{\theta^4}{4!} + \frac{\theta^6}{6!} - \cdots)C^2 \\
&= I + \sin\theta C + (1-\cos \theta)C^2
\end{align}


text:
# Solution

Show that if $a$ and $b$ are integers such that $a \mid b$, then $a^n \mid b^n$ for every positive integer $n$

If $a$ and $b$ are integers such that $a \mid b$, then $b = ak_1$ for some integer $k_1$. Now, raising both sides to the $n^\textrm{th}$ power, we have \begin{align}b^n &= (ak_1)^n\\ &= a^n {k_1}^n\end{align} However, ${k_1}^n$ must be an integer due to the closure of the integers under multiplication. So, defining $k_2 = {k_1}^n$, we see that $b^n = a^n k_2$ for some integer $k_2$. Hence, $a^n \mid b^n$.

Back


text:
My Math Forum HELP!! Maths problem has been bugging me for too long

 New Users Post up here and introduce yourself!

 April 22nd, 2015, 05:08 AM #1 Newbie   Joined: Apr 2015 From: birmingham Posts: 1 Thanks: 0 HELP!! Maths problem has been bugging me for too long Hello I am new to this forum and have been driven here by a problem that has been keeping me up for far too long now... If you were to take a sphere and slice through it at any point, except the center line, what would be the formula for the volume of the section of sphere you have sliced off? I have spent far too long trying to work out what seems to be a simple problem Thus far I have established the derivative of the volume of a sphere to get the gradient at any point on the remaining section but am not totally sure this is the right track to be heading down! PLEASE HELP!
 April 22nd, 2015, 07:00 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 Have you taken a Calculus class? That is the technique you would want to use. Setting up a coordinate system so that the origin is at the center of the sphere, you can write its equation as $\displaystyle x^2+ y^2+ z^2= R^2$ where R is the radius of the sphere. We can take the plane slicing the sphere to be $\displaystyle z= z_0$. That plane cuts the sphere in the circle $\displaystyle x^2+ y^2+ z_0^2= R^2$ or $\displaystyle x^2+ y^2= R^2- z_0^2$, a circle of radius $\displaystyle \sqrt{(R^2- z_0^2}$. That tells us that with x going from $\displaystyle -\sqrt{(R^2- z_0^2}$ to $\displaystyle \sqrt{(R^2- z_0^2}$, y goes from $\displaystyle -\sqrt{R^2- z_0^2- x^2}$ to $\displaystyle \sqrt{R^2- z_0^2- x^2}$. And, for each (x, y), the height will be $\displaystyle z= \sqrt{R^2- x^2- y^2}- y_0$. We can find the volume of that by $\displaystyle \int_{-\sqrt{(R^2- z_0^2}}^{\sqrt{(R^2- z_0^2}} \int_{-\sqrt{R^2- z_0^2- x^2}}^{\sqrt{R^2- z_0^2- x^2}}\left(\sqrt{R^2- x^2- y^2}- y_0\right) dy dx$
 April 22nd, 2015, 07:29 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2643 Math Focus: Mainly analysis and algebra Rather than bothering with spherical coordinates, I would look at the volume of revolution about the $x$-axis between $x = -a$ and $x = b$ (where $0 \lt b \lt a$) of the semicircle $f(x) = \sqrt{a^2 - y^2}$. Also, take a look here.

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text:
### Lecture Diary

08/23: Introduced quotient topology (can read about this in Munkres's book "Topology"). Discussed some basic examples, including taking quotients of spaces by group actions. Introduced real projective space, spheres, 2-torus, Mobius band, Klein bottle. Showed some basic relationships between them. For the basics of surfaces, I recommend taking a look at either Massey's book "Algebraic Topology: An Introduction" or Armstrong's book "Basic Topology" (both of which do way more than I will do).

08/25: Talked about surfaces some more. Also talked about graphs and the weak topology. All of this can be found in Munkres and Massey.

08/28: I introduced CW complexes. A nice source for what I covered (and more) is Chapter 0 of Hatcher's book (he alternates between calling them cell complexes and CW complexes; I will try to stick with calling them CW complexes). My lectures basically covered pages 5-8.

08/30: I introduced covering spaces, basically covering Chapter 2 of my (forever unfinished) book here.

09/01: I talked about lifts and covering space morphisms, basically covering Chapter 3 of my book.


text:
Tidy summarizes information about the components of a model. A model component might be a single term in a regression, a single hypothesis, a cluster, or a class. Exactly what tidy considers to be a model component varies across models but is usually self-evident. If a model has several distinct types of components, you will need to specify which components to return.

# S3 method for lm.beta
tidy(x, conf.int = FALSE, conf.level = 0.95, ...)

## Arguments

x An lm.beta object created by lm.beta::lm.beta. Logical indicating whether or not to include a confidence interval in the tidied output. Defaults to FALSE. The confidence level to use for the confidence interval if conf.int = TRUE. Must be strictly greater than 0 and less than 1. Defaults to 0.95, which corresponds to a 95 percent confidence interval. Additional arguments. Not used. Needed to match generic signature only. Cautionary note: Misspelled arguments will be absorbed in ..., where they will be ignored. If the misspelled argument has a default value, the default value will be used. For example, if you pass conf.lvel = 0.9, all computation will proceed using conf.level = 0.95. Additionally, if you pass newdata = my_tibble to an augment() method that does not accept a newdata argument, it will use the default value for the data argument.

## Details

If the linear model is an mlm object (multiple linear model), there is an additional column response.

If you have missing values in your model data, you may need to refit the model with na.action = na.exclude.

Other lm tidiers: augment.glm(), augment.lm(), glance.glm(), glance.lm(), glance.summary.lm(), glance.svyglm(), tidy.glm(), tidy.lm(), tidy.mlm(), tidy.summary.lm()

## Value

A tibble::tibble() with columns:

conf.high

Upper bound on the confidence interval for the estimate.

conf.low

Lower bound on the confidence interval for the estimate.

estimate

The estimated value of the regression term.

p.value

The two-sided p-value associated with the observed statistic.

statistic

The value of a T-statistic to use in a hypothesis that the regression term is non-zero.

std.error

The standard error of the regression term.

term

The name of the regression term.

## Examples

library(lm.beta)

mod <- stats::lm(speed ~ ., data = cars)
std <- lm.beta(mod)
tidy(std, conf.int = TRUE)
#> # A tibble: 2 × 8
#>   term        estimate std_estimate std.error statistic  p.value conf.low conf.high
#>   <chr>          <dbl>        <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
#> 1 (Intercept)    8.28         0        0.874       9.47 1.44e-12   -1.76      1.76
#> 2 dist           0.166        0.807    0.0175      9.46 1.49e-12    0.772     0.842
ctl <- c(4.17, 5.58, 5.18, 6.11, 4.50, 4.61, 5.17, 4.53, 5.33, 5.14)
trt <- c(4.81, 4.17, 4.41, 3.59, 5.87, 3.83, 6.03, 4.89, 4.32, 4.69)
group <- gl(2, 10, 20, labels = c("Ctl", "Trt"))
weight <- c(ctl, trt)

mod2 <- lm(weight ~ group)

std2 <- lm.beta(mod2)
tidy(std2, conf.int = TRUE)
#> # A tibble: 2 × 8
#>   term        estimate std_estimate std.error statistic  p.value conf.low conf.high
#>   <chr>          <dbl>        <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
#> 1 (Intercept)    5.03         0         0.220     22.9  9.55e-15   -0.463     0.463
#> 2 groupTrt      -0.371       -0.270     0.311     -1.19 2.49e- 1   -0.925     0.384


text:
# Proof sin pi/6

1. Aug 18, 2011

### leitz

I'm currently reading Apostol's Calculus. I'm now on the part where I need to prove sin and cos values for specific values. Right now I'm working on sin and cos of pi/6. However, I always get at 1/2 and √3/2 for the final value. Here's my solution so far:

1. The problem statement, all variables and given/known data
Use the properties of sine and cosine to prove that sin pi/6 =1/2. (It must be analytical, not geometrical)

2. Relevant equations
The book said that I could use exercise 4, which was to prove that sin3x= 3 sinx - 4sin3x , cos3x = cos x - 4sin2x cosx.

3. The attempt at a solution
sin 2x = 2sinxcosx
sin 3x = 3sinx - 4sin3x

=> sin 6x = 2sin3x cos3x = 2sin3x√(1-sin23x)

=>sin 6x = 2(3sinx - 4sin3x)√[1-(3sinx - 4sin3x)2]

let x = pi/6
=> sin pi = 2(3sin(pi/6) - 4sin3(pi/6))√[1-(3sin(pi/6) - 4sin3(pi/6))2] = 0 (Using the property of sine - sin pi = sin 0 = 0)

By using the zero factor theorem (Theorem I.11 on Apostol's book), either
(3sin(pi/6) - 4sin3(pi/6))=0 or √[1-(3sin(pi/6) - 4sin3(pi/6))2]=0

After simplifying the equations, it will turn out that sin pi/6 is equal to 0, √3/2 , -√3/2 , -1, and 1. But since pi/6 is in the above the real axis and is in the first quadrant, then the value of sin pi/6 MUST be positive, eliminating the negative roots. Furthermore, since it was also established as a property in the book that sin 0 = 0 and sin pi/2 =1, then we are only left with √3/2 and 1/2.

This is where my problem lies. Any idea how to continue? Also, What would be a nice start for the cosine counter part of it?

Last edited: Aug 19, 2011
2. Aug 19, 2011

### PeterO

When you did this line:

=> sin 6x = 2sin3x cosx = 2sin3x√(1-sin23x)

why was it not 2 sin3x.con3x ??

3. Aug 19, 2011

### leitz

sorry 'bout that typo..

4. Aug 19, 2011

### dynamicsolo

You still have the equation for, say, cos 3x. Presumably, you also have the Pythagorean Identity, so you know that cos(pi/2) = 0 , since sin(pi/2) = 1 . So we have

cos(pi/6) [ 1 - 4 sin2(pi/6) ] = 0 .

If sin(pi/6) = 1/2 , then cos(pi/6) = (√3)/2 , and vice versa [also by Pythagorean Identity]. Which of these possibilities solves the equation?

5. Aug 19, 2011


text:
# Difference between revisions of "2001 AMC 10 Problems/Problem 7"

## Problem

When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?

$\textbf{(A) }0.0002\qquad\textbf{(B) }0.002\qquad\textbf{(C) }0.02\qquad\textbf{(D) }0.2\qquad\textbf{(E) }2$

## Solution

We can write our equation as:

$\frac{x}{10000}=\frac{4}{x}$

Cross-multiply and solve for $x$.

$x^2=40000 \implies x=200$.

$200/10000=2/100= \boxed{\textbf{(C) }0.02}$.


text:
# Trigonometry (Law of Cosines)

## Recommended Posts

In case of right angled triangles, C^2 = A^2 +B^2 - 2AB cos(Ψ) is shortened to C^2 = A^2 +B^2 because the cosine of the angle "Ψ" which is 90° is equal to 0. But how is its cosine equal to 0.

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This illustration of sine and cosine curves might help: https://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html

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1 minute ago, Strange said:

This illustration of sine and cosine curves might help: https://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html

I know that but they don't answer my question.

##### Share on other sites

One way of defining cosine is as proportional to the horizontal projection of what you call B in your figure. The segment B would be the one that sweeps the angle. That's why it's zero.

The sine of 90º is 1, on the contrary, because the vertical projection of B is just B.

1 hour ago, King E said:

I know that but they don't answer my question.

The post does answer your question, although it may be difficult to see because it's moving. Take a look at the horizontal projection of the triangle when it goes through 90º.

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2 hours ago, King E said:

In case of right angled triangles, C^2 = A^2 +B^2 - 2AB cos(Ψ) is shortened to C^2 = A^2 +B^2 because the cosine of the angle "Ψ" which is 90° is equal to 0. But how is its cosine equal to 0.

Did you really learn about the cosine rule before you learned a definition of the cosine?

What is your definition of a cosine?

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2 hours ago, King E said:

I know that but they don't answer my question.

Can you explain why not?  Or ask your question more clearly

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Posted (edited)
27 minutes ago, Strange said:

50 minutes ago, studiot said:

27 minutes ago, Strange said:

Can you explain why not?  Or ask your question more clearly

Yes because you find faults in the question whenever your answer doesn't help.

Edited by King E

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18 minutes ago, King E said:

Yes because you find faults in the question whenever your answer doesn't help.

Uh, OK.

Did this answer help more:

1 hour ago, joigus said:

The sine of 90º is 1, on the contrary, because the vertical projection of B is just B.

(I thought it was spot on.)

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1 minute ago, Strange said:

Uh, OK.

Did this answer help more:

(I thought it was spot on.)

Yes it helped and I got my answer.

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Posted (edited)
7 hours ago, King E said:

Yes it helped and I got my answer.

However I'm disappointed with your response to my earlier question, which I asked for a particular reason, I had intended to develop for your benefit.

In my first encounter with cosines I was given this definition, which referred to right angled triangles only.

$\cos \psi = \frac{{adjacent}}{{hypotenuse}}$

Using this definition can you see that there is a problem with this, in respect of your question ?

Since c has to be the hypotenuse, which of the two sides corresponds to the adjacent side ?  a or b ?

Clearly there are two possible answers here and the solution to this problem is geometrically interesting.

It is very important to get this straight in your head because both Pythagoras and the Cosine rule are deeply embedded in fundamental Physics.

Edited by studiot

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Another approach cos(x)=sin(90-x), so cos(90)=sin(0).  In a right triangle, it is obvious sin(0)=0.

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text:
# Math Help - Integral

1. ## Integral

A bacteria population starts with 400 bacteria and grows at a rate of
r(t)= (450.268) e^1.12567t bacteria per hour.
How many bacteria will there be after 3 hours?

2. Originally Posted by camherokid
A bacteria population starts with 400 bacteria and grows at a rate of
r(t)= (450.268) e^1.12567t bacteria per hour.
How many bacteria will there be after 3 hours?
For this, you need only to plug in 3 for t:

r(3) = (450.268) e^(1.12567*3) = 13185.2355160

Round this number to as many digits as you need for you solution.

3. Originally Posted by camherokid
A bacteria population starts with 400 bacteria and grows at a rate of
r(t)= (450.268) e^1.12567t bacteria per hour.
How many bacteria will there be after 3 hours?
How can the rate of growth be
r(t) = (450.268) e^(1.12567t)

The coefficient (450.268) of the exponential function (e^(1.12567t)) should be equal to the initial population (400).
I don't see why you would have 450.268 in front of the e.

4. Originally Posted by ecMathGeek
For this, you need only to plug in 3 for t:

r(3) = (450.268) e^(1.12567*3) = 13185.2355160

Round this number to as many digits as you need for you solution.
Oops... I just noticed this is the growth rate and not the population at time t.

r(t) = (450.268) e^(1.12567*t)

P(t) = INT r(t) dt = INT (450.268) e^(1.12567*t) dt
P(t) = (450.268/1.12567)e^(1.12567*t) + C

At t = 0, P(t) = 400
400 = (400)e^(1.12567*0) + C --> C = 0

P(t) = (400)e^(1.12567*t)

Solving for the population at t = 3, we get
P(3) = (400)e^(1.12567*3) = 11713.2334663