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Course

Subject:	Mathematics
Topic:	Linear Algebra
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Name:	Math Doctor Bob (Robert Donley)
Type:	Individual

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Linear Algebra I: From Linear Equations to Eigenspaces

Video Lectures

Displaying all 47 video lectures.

Lecture 1 Play Video	Row Reduction for a System of Two Linear Equations Linear Algebra: Find all solutions of the system of linear equations 2x - 3y = 2, -x + 2y = 1 using row reduction. We apply row operations to an augmented matrix.
Lecture 2 Play Video	Solving a 2x2 SLE Using a Matrix Inverse Linear Algebra: Solve the system of linear equations: 5x + 3y = 4, 2x + 2y = 1 using the inverse of a 2 x 2 matrix.
Lecture 3 Play Video	Solving a SLE in 3 Variables with Row Operations 1 Linear Algebra: We solve a system of three linear equations in three variables using row operations. In this Part, we give a procedure for row reduction and give an example using coefficients of 0 and 1.
Lecture 4 Play Video	Solving a SLE in 3 Variables with Row Operations 2 Linear Algebra: We solve a system of linear equations in 3 variables by performing row operations on an augmented matrix. In Part 1, we use equations with 0-1 coefficients; here we consider general integer coefficients.
Lecture 5 Play Video	Consistency of a System of Linear Equations Linear Algebra: Determine whether the following systems of linear equations are consistent: (a) x -3y +2z + 2w = 1, 2x - 2z = 3, 4x - 6y +2z + 4w = 6; (b) x-3y+2z+2w=1, 2x-2z = 3, 4x - 6y - 2z + 4w = 5.
Lecture 6 Play Video	Inverse of 3 x 3 Matrix Using Row Operations 1 Linear Algebra: Find the inverse of the 3 x 3 matrix [ 1 1 1 \ 1 0 1\ 1 0 0 ] by using row operations. We check our answer using the equation AA^{-1} = I. In Part 2, we consider a matrix with general integer coefficients.
Lecture 7 Play Video	Inverse of 3x3 Matrix Using Row Operations 2 Linear Algebra: We compute the inverse of a 3x3 matrix A by applying row operations to the matrix A augmented by the identity matrix. In Part 1, we worked this problem out for a 0-1 matrix; here we use general integers.
Lecture 8 Play Video	Inverse of 4x4 Matrix Using Row Operations Linear Algebra: We find the inverse of a real 4x4 matrix using row operations. We note the bookkeeping pattern and check the answer with the equation A^-1 A = I.
Lecture 9 Play Video	Example of Determinant Using Row Echelon Form Linear Algebra: Find the determinant of the 3 x 3 matrix A = [ 3 5 2 \ 2 2 4 \ 0 3 5] by using row operations to put A in row echelon form. We review the effect of row operations on determinants.
Lecture 10 Play Video	Inverse of 3 x 3 Matrix Using Adjugate Formula Linear Algebra: Find the inverse of the 3 x 3 matrix A = [ \ \ ] using the adjugate (or classical adjoint) of A. This is mostly a bookkeeping exercise.
Lecture 11 Play Video	Inverse of 4x4 Matrix Using Adjugate Formula Linear Algebra: We find the inverse of a 4x4 matrix using the adjugate (or classical adjoint) formula. Key steps include computing minors and the trick for 3x3 determinants.
Lecture 12 Play Video	Cramer's Rule for Three Linear Equations Linear Algebra: Using Cramer's Rule, find all solutions to the system of linear equations x+y = 3, 2x +y+z=2, -y=4.
Lecture 13 Play Video	Determinant of a 4 x 4 Matrix Using Cofactors Linear Algebra: Find the determinant of the 4 x 4 matrix A = [1 2 1 0 \ 2 1 1 1 \ -1 2 1 -1 \ 1 1 1 2] using a cofactor expansion down column 2. This is largely an exercise in bookkeeping.
Lecture 14 Play Video	Determinant of a 4 x 4 Matrix Using Row Operations Linear Algebra: Is the 4 x 4 matrix A = [ 1 2 1 0 \ 2 1 1 1 \ -1 2 1 -1 \ 1 1 1 2] invertible? We test invertibility by checking the determinant. We compute the determinant by performing row operations before using cofactors.
Lecture 15 Play Video	Examples of Linear Maps Linear Algebra: Here are a few problems on linear maps. Part 1: Are the following maps L:R^3 to R^3 linear? (a) L(x, y, z) = (x+1, x-y-2, y-z), (b) L(x, y, z) = (x + 2y, x-y-2z, 0). Part 2: Suppose L:R^3 to R^2 is linear and defined on the standard basis by L(e1) = (1, 2), L(e2) = (0, 3), and L(e3) = (1, -1). Compute L(2,-1,3). For both parts, we explain linearity in terms of linear combinations and in terms of matrix-based maps.
Lecture 16 Play Video	Example of Linear Combination Linear Algebra: Let v1 = [1 0 1 0], v2 = [2 3 0 0], v3 = [0 1 0 1]. Is the vector b = [-1 0 3 6] is the span of v1, v2, v3? If so, write b as a linear combination of v1, v2, and v3.
Lecture 17 Play Video	Example of Linear Combination (Visual) Linear Algebra: For the vectors u and v in the plane, express the vectors w1, w2, and w3 as linear combinations. We interpret linear combinations in terms of uv and xy coordinates.
Lecture 18 Play Video	Evaluating Linear Transformations Using a Basis Linear Algebra: Suppose L is a linear map from R^2 to R^2 such that L[1 -1] = [1 3] and L[2 2] = [0 -2]. Compute L[3 4] by writing [3 4] as a linear combination of [1 -1] and [2 2].
Lecture 19 Play Video	Linear Transformations on R^2 Linear Algebra: Let T be a linear transformation from R^2 to R^2. We consider various example for T and interpret geometrically. Examples include diagonal transformation and rotations.
Lecture 20 Play Video	Example of Checking for Basis Property Linear Algebra: Let B = {[1 0 -2], [3 2 -4], [-3 -5 1]}. Is B a basis for R^3? Let A be the matrix with columns formed from B. We consider the pivots in the row echelon form of A, and check our answer by computing the determinant of A.
Lecture 21 Play Video	Example of Basis for a Null Space Linear Algebra: Find a basis for the null space of the matrix A = [ 1 0 3 2 1 \ 0 2 2 4 4 \ 0 0 0 2 6 ]. We use reduced row echelon form to assign dependent and independent variables.
Lecture 22 Play Video	Example of Basis for a Span Linear Algebra: Find a basis for the span of the vectors {[1 2 3], [1 1 0], [5 8 9], [3 3 0]}. We present two approaches, one computational and one direct.
Lecture 23 Play Video	Example of Linear Independence Using Determinant Linear Algebra: Let S = {[12, 0, 4, 0], [3,1 , 1, 1], [3, 0, 2, 0], [3, 2, 0, 0]}. Show that S is a linearly independent set by computing the determinant of the matrix whose columns are the vectors in S.
Lecture 24 Play Video	Example of Kernel and Range of Linear Transformation Linear Algebra: Find bases for the kernel and range for the linear transformation T:R^3 to R^2 defined by T(x1, x2, x3) = (x1+x2, -2x1+x2-x3). We solve by finding the corresponding 2 x 3 matrix A, and find its null space and column span. We check our work using the Rank Equation.
Lecture 25 Play Video	Linear Transformations: One-One Linear Algebra: We recall the definition of one-one for functions and apply it to linear transformations. We obtain a simple rule for checking one-one in this case: either the kernel is zero or the associated matrix has a pivot in each column in row echelon form. Several examples are given.
Lecture 26 Play Video	Linear Transformations: Onto Linear Algebra: Continuing with function properties of linear transformations, we recall the definition of an onto function and give a rule for onto linear transformations.
Lecture 27 Play Video	Example of Change of Basis Linear Algebra: Consider the bases for R^2: B = {[1 1], [0 2]} and C={[0 1], [1 2]}. If the coordinate vector for x with respect to B is [3 1], find the coordinate vector for x with respect to C.
Lecture 28 Play Video	Eigenvalues and Eigenvectors Linear Algebra: We introduce eigenvalues and eigenvectors by considering the equation Av=cv. After considering the geometric content of this equation, we provide a procedure for finding eigenvalues and eigenvectors and apply it to the matrix A = [ 1 1 0 / 1 0 1/ 0 1 1 ].
Lecture 29 Play Video	Example of Eigenvector: Markov Chain Linear Algebra: A bird population migrates between Canada, USA, and Mexico. Each year, the migration pattern is as follows: for the Canadian birds, 30% go to the USA and 30% go to Mexico; for the USA birds, 50% go to Canada and 0% got to Mexico; for the Mexican birds, 10% go to Canada and 0% got to the USA. Find the transition matrix of the Markov chain and a stable solution.
Lecture 30 Play Video	Example of Diagonalizing a 2 x 2 Matrix Linear Algebra: Let A = [3 1\ -2 0]. Find a 2 x 2 matrix P and a diagonal 2 x 2 matrix D such that P^{-1}AP = D.
Lecture 31 Play Video	Example of Power Formula for a Matrix Linear Algebra: Find the equation for A^n where A = [2 -1 0 \ 0 1 0 \ -3 -2 -1]. We put A in diagonal form by finding a basis of eigenvectors. The formula follows from the simpler rule for a power of a diagonal matrix. We verify the formula through A^3 by hand.
Lecture 32 Play Video	The Fibonacci Numbers Using Linear Algebra (HD Version) Linear Algebra: We derive the Binet Formula for the Fibonacci numbers using linear algebra. The technique involves using diagonalization to compute the power of a matrix.
Lecture 33 Play Video	Vector Length in R^n Linear Algebra: Using the standard length of vectors in R, R^2, and R^3, we show how nonzero vectors factor into length and direction. The direction is represented by a unit vector. A general formula is given for R^n using the Pythagorean Theorem.
Lecture 34 Play Video	The Standard Inner Product on R^n Linear Algebra: We define the standard inner product on R^n and explain its basic properties. A cosine formula is given in terms of the inner product and lengths of two vectors.
Lecture 35 Play Video	Example of Fourier's Trick Linear Algebra: Given an orthonormal basis of R^n, we present a quick method for finding coefficients of linear combination in terms of the basis. We also give an analogue of Parseval's Identity, which relates these coefficients to the squared length of the vector.
Lecture 36 Play Video	Example of Orthogonal Complement Linear Algebra: Let u = (1, 2, -1) in R^3, and let W be the subspace of all vectors in R^3 orthogonal to u. Find a basis of unit vectors for W.
Lecture 37 Play Video	Orthogonal Transformations 1: 2x2 Case Linear Algebra: Let A be a 2x2 orthogonal matrix. A general form for A is given, and we show that A corresponds to either a rotation or reflection of the plane. (Added: Minor edit to reflections.)
Lecture 38 Play Video	Orthogonal Transformations 2: 3x3 Case Linear Algebra: Let A be a 3x3 orthogonal matrix. We describe A as a rotation of R^3 about some line through the origin and give a recipe for finding the angle in terms of det(A) and Trace(A). An explicit example is given.
Lecture 39 Play Video	Example of Gram-Schmidt Orthogonalization Linear Algebra: Construct an orthonormal basis of R^3 by applying the Gram-Schmidt orthogonalization process to (1, 1, 1), (1, 0, 1), and (1, 1, 0). In addition, we show how the Gram-Schmidt equations allow one to factor an invertible matrix into an orthogonal matrix times an upper triangular matrix.
Lecture 40 Play Video	QR-Decomposition for a 2x2 Matrix Linear Algebra: We give a general formula for a QR-decomposition of a real 2x2 matrix; that is, we show how to decompose any 2x2 matrix A as a product QR where Q is orthogonal and R is upper triangular. We also note one set of conditions under which the factorization is unique.
Lecture 41 Play Video	Beyond Eigenspaces: Real Invariant Planes Linear Algebra: In the context of real vector spaces, one often needs to work with complex eigenvalues. Let A be a real nxn matrix A. We show that, in R^n, there exists at least one of: an (nonzero) eigenvector for A, or a 2-dimensional subspace (plane) invariant under A.
Lecture 42 Play Video	Beyond Eigenspaces 2: Complex Form Linear Algebra: As an application of the Spectral Theorem for real vector spaces, we show that every 2x2 matrix with no real eigenvalues can be represented as [x -y / y x] for some basis. This representation reflects common algebraic properties of the complex numbers.
Lecture 43 Play Video	Spectral Theorem for Real Matrices: General 2x2 Case Linear Algebra: We state and prove the Spectral Theorem for a real 2x2 symmetric matrix A = [a b \ b c]. That is, we show that the eigenvalues of A are real and that there exists an orthonormal basis of eigenvectors for A.
Lecture 44 Play Video	Spectral Theorem for Real Matrices: General nxn Case Linear Algebra: We state and prove the Spectral Theorem for real vector spaces. That is, if A is a real nxn symmetric matrix, we show that A can be diagonalized using an orthogonal matrix. The proof refers to the 2x2 case and to results from the video Beyond Eigenspaces: Real Invariant Planes.
Lecture 45 Play Video	Example of Spectral Theorem (3x3 Symmetric Matrix) Linear Algebra: We verify the Spectral Theorem for the 3x3 real symmetric matrix A = [ 0 1 1 / 1 0 1 / 1 1 0 ]. That is, we show that the eigenvalues of A are real and that there exists an orthonormal basis of eigenvectors. In other words, we can put A in real diagonal form using an orthogonal matrix P. (Eigenvalues and eigenvectors for this A are found in the video "Eigenvalues and Eigenvectors.")
Lecture 46 Play Video	Example of Spectral Decomposition Linear Algebra: Let A be the real symmetric matrix [ 1 1 4 / 1 1 4 / 4 4 -2 ]. Using the Spectral Theorem, we write A in terms of eigenvalues and orthogonal projections onto eigenspaces. Then we use the orthogonal projections to compute bases for the eigenspaces.
Lecture 47 Play Video	Example of Diagonalizing a Symmetric Matrix (Spectral Theorem) Linear Algebra: For the real symmetric matrix [3 2 / 2 3], 1) verify that all eigenvalues are real, 2) show that eigenvectors for distinct eigenvalues are orthogonal with respect to the standard inner product, and 3) find an orthogonal matrix P such that P^{-1}AP = D is diagonal. The Spectral Theorem states that every symmetric matrix can be put into real diagonal form using an orthogonal change of basis matrix (or there is an orthonormal basis of eigenvectors).