**Math 4A**

Linear Algebra with Applications

- PatrickJMT: Linear System of Equations: Row Reducing - Part 1
Video - 9:59 Introduction to row reduction Continued in part 2 - PatrickJMT: Using Gauss-Jordan to Solve a System of Three Linear Equations - Example 1
Video - 7:12 Example of solving a linear system (unique solution) - Khan Academy: Matrices: RREF 1
Video - 17:43 Solves a linear system that has an infinite solution set See following 2 videos for more examples - PurpleMath: Matrix Row Operations
Text Description and examples of matrix (elementary) row operations - LinearAlgebraToolkit: Row Operation Calculator
Interactive Allows you to enter in a matrix and then select which row operations you wish to perform Good way to check whether your work for arithmetic errors - LinearAlgebraToolkit: Transforming a Matrix to Reduced Row Echelon Form
Interactive Allows you to enter in a matrix and then shows step-by-step which row operations are used to put matrix into RREF - LinearAlgebraToolkit: Solving a System of Linear Equations
Interactive Allows you to enter in an augmented matrix and then shows step-by-step which row operations are used to solve the system of equations

- Khan Academy: Introduction to Vectors
Video - 16:31 Lecture on vectors - Khan Academy: Vectors
Interactive Exercises on vectors - PatrickJMT: An Introduction to Vectors
Video - 4:46 Introduction to vectors and magnitude

- Khan Academy: Linear Combinations and Span
Video - 20:34 Lecture on span - Wolfram|Alpha
Interactive Do-it-all mathematics tool Use Span (Ex: Span{(1,2),(3,4)}) to calculate the span of an entered set of vectors - LinearAlgebraToolkit: Determining if a Set Spans a Space
Interactive Allows you to enter in a set of vectors of R^n, P^n, or M^n, and shows how to determine whether the set spans the space - PatrickJMT: The Span of a Set of Vectors
Video - 17:06 Examples of finding the span of a set of vectors

- Khan Academy: Introduction to Linear Independence
Video - 15:46 Lecture on linear independence See following two videos for more formal definition and examples - Wolfram|Alpha
Interactive Do-it-all mathematics tool Use Linear Independence (Ex: Linear Indepence {(1,2),(3,4)}) to determine whether entered vectors are linearly independent - LinearAlgebraToolkit: Linear Independence and Dependence
Interactive Allows you to enter in a set of vectors of R^n, P^n, or M^n, and shows how to determine linear independence - PatrickJMT: Linear Independence and Linear Dependence, Ex. 1
Video - 8:05 Examples on determining whether a set of vectors is linearly independent

- Khan Academy: Linear Transformations
Video - 13:52 Lecture on linear transformations - Khan Academy: Linear Transformations as Matrix Vector Products
Video - 17:32 Determining the matrix of a linear transformation given how it acts on basis vectors - Pat Plunkett Math Grad: 2 x 2 Matrices
Interactive Visualize how a linear transformation corresponding to a 2x2 matrix affects a pentagon - PatrickJMT: Linear Transformations, Example 1, Part 1 of 2
Video - 9:06 Definition of a linear transformation and then some examples

- Khan Academy: Matrix Product Examples
Video - 18:14 Example of a matrix product - PatrickJMT: Basic Matrix Operations
Video - 5:25 Matrix addition, subtraction, scalar multiplication - PatrickJMT: Multiplying a Matrix by another Matrix
Video - 9:38 Example of a matrix product

- Khan Academy: Example of Finding Matrix Inverse
Video - 6:22 Example of finding a matrix inverse - Wolfram|Alpha
Interactive Do-it-all mathematics tool Enter in matrix (ex: {{1,2,3},{3,2,1},{2,1,3}}) and will find inverse - LinearAlgebraToolkit: Calculating the Inverse using Row Operations
Interactive Allows you to enter in a square matrix and then will show step-by-step how to find its inverse - PurpleMath: Matrix Inversions
Text Description and examples for finding the inverse of a matrix

- Khan Academy: 3x3 Determinant
Video - 10:01 Example of finding a 3x3 determinant - PatrickJMT: Finding the Determinant of a 3x3 Matrix
Video - 6:56 How to find a 3x3 determinant - Wolfram|Alpha
Interactive Do-it-all mathematics tool Enter in matrix (ex: {{1,2,3},{3,2,1},{2,1,3}}) and will calculate its determinant - PurpleMath: Minors and Cofactors
Text Description on taking determinants using cofactor expansion, and by using row operations

- Khan Academy: Linear Subspaces
Video - 23:39 Lecture on subspaces of R^n - Khan Academy: Introduction to the Null Space of a Matrix
Video - 10:23 Lecture on the null space of a matrix - Khan Academy: Column Space of a Matrix
Video - 10:40 Lecture on the column space of a matrix

- Khan Academy: Basis of a Subspace
Video - 19:00 Lecture on bases - LinearAlgebraToolkit: Finding the Basis of a Set
Interactive Allows you to enter in a set of vectors of R^n, P^n, or M^n, and determines if the set is a basis - PatrickJMT: Procedure to Find a Basis for a Set of Vectors
Video - 7:17 Examples on finding the basis of a set of vectors - Khan Academy: Coordinates with Respect to a Basis
Video - 16:06 Lecture on coordinate systems - Khan Academy: Dimension of the Column Space or Rank
Video - 12:48 Lecture on rank

- Khan Academy: Introduction to Eigenvalues and Eigenvectors
Video - 7:43 Lecture on eigenvectors and eigenvalues - Wolfram|Alpha
Interactive Do-it-all mathematics tool Enter in matrix (ex: {{1,2,3},{3,2,1},{2,1,3}}) and will calculate its eigenvalues/eigenvectors - MIT: Eigenvalues Lecture 1
Video - Slides with lecture on characteristic equation and finding eigenvalues - MIT: Eigenvalues Lecture 2
Video - Slides with lecture on finding eigenvectors - MIT: Eigenvalue Demonstration
Interactive Interactive applet that calculates eigenvalues/eigenvectors and plots the image of the unit circle under a transformation - Khan Academy: Showing that an Eigenbasis makes for good coordinate systems
Video - 13:09 Lecture on diagonalization - MIT: Eigenvalues Lecture 4
Video - Slides with lecture on diagonalization

- Khan Academy: Projections onto Subspaces
Video - 17:26 Lecture on projections - Khan Academy: Introduction to Orthonormal Bases
Video - 11:16 Lecture on orthonormal bases - Khan Academy: Least Squares Approximation
Video - 15:32 Lecture on least squares approximation Next two videos give examples

- Applications of Linear Algebra
Text List and descriptions of applications of linear algebra - NYTimes: College Football Computer Rankings Proliferate
Text On the increasing popularity of math-based sports rankings - AMS Feature: How Google Finds Your Needle in the Web's Haystack
Text Background behind Google's PageRank algorithm - The $25,000,000,000 Eigenvector
Text For the adventurous student - a more detailed look at how linear algebra powers Google's PageRank algorithm