Eigenvectors and eigenvalues explained visually Explained Visually By Victor Powell and Lewis Lehe Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's PageRank algorithm. To begin, let $v$ be a vector (shown as a point) and $A$ be a matrix with columns $a_1$ and $a_2$ (shown as arrows). If you can draw a line through the three points $(0,0)$, $v$ and $Av$, then $Av$ is just $v$ multiplied by a number $\lambda$; that is, $Av = \lambda v$. Below, change the columns of $A$ and drag $v$ to be an eigenvector. What are eigenvalues/vectors good for? If you keep multiplying $v$ by $A$, you get a sequence ${ v, Av, A^2v,}$ etc. Let's explore some applications and properties of these sequences. Fibonacci Sequence Suppose you have some amoebas in a petri dish. which we can rewrite in matrix form like Below, press "Forward" to step ahead a minute. 1 child + 0 adults = 1 As you can see, the system goes toward the grey line, which is an eigenspace with $\lambda = (1+\sqrt 5)/2 > 1$.
Yudkowsky - Bayes' Theorem An Intuitive Explanation of Bayes' Theorem Bayes' Theorem for the curious and bewildered; an excruciatingly gentle introduction. Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. It's this equation. So you came here. Why does a mathematical concept generate this strange enthusiasm in its students? Soon you will know. While there are a few existing online explanations of Bayes' Theorem, my experience with trying to introduce people to Bayesian reasoning is that the existing online explanations are too abstract. Or so they claim. And let's begin. Here's a story problem about a situation that doctors often encounter: 1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. What do you think the answer is? Group 1: 100 women with breast cancer.
Bayes’ Theorem, Predictions and Confidence Intervals / Algorithms There are plenty of articles on this subject, but they do not review real-life problems. I am going to try to fix this. We use Bayes’ Theorem (a.k.a. Bayes’ law or Bayes’ rule) to filter spam in recommendation services and for ratings system. A great number of algorithms of fuzzy searches would be impossible without it. Besides, the theorem is the cause of holy wars between mathematicians. Introduction Let’s start from the very beginning. Let’s introduce the following notions: P(A) is the probability of A event occurrence.P(AB) is the probability of both events occurrence. Let’s think. It’s so simple that 300 years ago a priest derived it. Direct and Inverse Problems We can describe a direct problem the following way: find the probability of one of effects by a reason. But the problem is that we do not know the probability of some event in minority of cases only. In practice, the inverse task is more important. We have to use some function. Binomial Distribution P([s,d]|p=x). Holy War
Differential Equations Explained $\cos$PLAY You're probably used to equations like $$(t-.5)(t-1)= 0,$$ where 'solving' means finding an unknown number. A differential equation (DE), by contrast, is a fact about the derivative of an unknown function, and 'solving' one means finding a function that fits. To visualize derivatives, we can draw a right triangle whose hypoteneuse is tangent to a function. If the triangle's width is $1$, then its height is the derivative. With that one weird trick, the plots to the right show how the derivative of $\sin(t)$ is $\cos(t)$. That's a pretty basic DE, though. Consider a cart rolling to a stop. The solution is a function $v(t)$ giving velocity at time $t$. It turns out the exponential function, $e^{-kt}$, has the properties $$ \begin{align} \frac{d}{dt}e^{-kt}=-ke^{-kt} && e^{-k\cdot 0}=1. To make the solution more intuitive, here you'll solve the cart's DE manually by picking a series of $\left( t, v \right)$ points. The first cart below obeys the $v(t)$ function you designed.
Immersive Linear Algebra immersivemath immersive linear algebra by J. Ström, K. Åström, and T. The world's first linear algebra book with fully interactive figures. Learn More Check us out on Twitter and Facebook Table of Contents Preface A few words about this book. Chapter 1: Introduction How to navigate, notation, and a recap of some math that we think you already know. Chapter 2: Vectors The concept of a vector is introduced, and we learn how to add and subtract vectors, and more. Chapter 3: The Dot Product A powerful tool that takes two vectors and produces a scalar. Chapter 4: The Vector Product In three-dimensional spaces you can produce a vector from two other vectors using this tool. Chapter 5: Gaussian Elimination A way to solve systems of linear equations. Chapter 6: The Matrix Enter the matrix. Chapter 7: Determinants A fundamental property of square matrices. Chapter 8: Rank Discover the behaviour of matrices. Chapter 9: Linear Mappings Learn to harness the power of linearity... Chapter 10: Eigenvalues and Eigenvectors
An Interactive Guide To The Fourier Transform The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations: Yikes. Rather than jumping into the symbols, let's experience the key idea firsthand. What does the Fourier Transform do? Here's the "math English" version of the above: The Fourier Transform takes a time-based pattern, measures every possible cycle, and returns the overall "cycle recipe" (the strength, offset, & rotation speed for every cycle that was found). Time for the equations? If all goes well, we'll have an aha! This isn't a force-march through the equations, it's the casual stroll I wish I had. From Smoothie to Recipe A math transformation is a change of perspective. The Fourier Transform changes our perspective from consumer to producer, turning What did I see? In other words: given a smoothie, let's find the recipe. Why? So... given a smoothie, how do we find the recipe? Well, imagine you had a few filters lying around: Filters must be independent. Whoa. Stop.
Math ∩ Programming