**Lecture Schedule**

1/9 Introduction and motivation, 4 ways to view matrix multiplication, upper/lower triangular (UT/LT) matrices, solving Ax=b for a triangular matrix, Gauss Elimination

1/11 How Gauss Elimination works, Gauss Elimination via matrix multiplication, pivots, permutation matrices, PA = LU decomposition, computational cost of LU decomposition, PA=AU via Matlab, inverses

1/16 Matrix inverse via Gauss-Jordan method, linear regression, transpose, symmetric matrices, vector spaces

1/18 Vector spaces and subspaces, matrix groups, 4 fundamental subspaces associated to a matrix, N(A) & C(A) and relations to solving Ax=b

1/23 Echelon form and reduced echelon form, finding N(A), solving Ax=b, numerical solution of boundary value problem

1/25 More about linear independence, basis, dimensions of N(A) and C(A)

1/30 Fundamental Theorem of Linear Algebra I and corollaries, L/R inverses of mxn matrices

2/1 Numerical issues inverting Hilbert matrix, more about L/R inverses, polynomial interpolation, graphs and incidence matrices, rank-1 matrices, linear transformations, transformations of the plane and computer graphics, vector spaces of polynomials, differentiation via matrix multiplication

2/6 Fundamental Theorem of Calculus for polynomials, inner products, orthogonality, Fundamental Theorem of Orthogonality

2/8 Orthogonal complement, Fundamental Theorem of Linear Algebra II, Fredholm's alternative, orthogonal projection

2/13 Adjoint, Cauchy-Schwarz inequality, orthogonal projection onto a subspace, best approximate solution of Ax=b, projection matrices

2/15 Linear regression revisited, orthonormal bases, orthogonal matrices, Gram-Schmidt orthogonalization

2/20 Exam 1

2/22 Legendre polynomials, Fourier series, A=QR decomposition, application of A=QR to least squares, intersections of subspaces

2/27 Sum of subspaces, direct sum of subspaces, cartesian product of subspaces, isomorphism of vector spaces, tensor product of spaces, definition of determinant via 3 properties

3/1 Properties of determinants, formula for det, existience and unqiueness of det, permutations, det(AB)=det(A)det(B), det of triangular matrix, determinant of any square matrix using PA=PDU, det(A^T)=det(A), determinant via co-factors

3/6 More about co-factors, matrix inverse via co-factors, geometric interpretation of determinants, eigenvalues and eigenvectors, diagonalization

3/8 Solution of linear systems of ODEs x'=Ax, computing matrix powers, Leslie population model, Google's PageRank algorithm

3/13 Perron-Frobenius theorem (statement), square root of a matrix, brief review of complex numbers, complex matrices, Hermitian matrices, spectral properties of symmetric and Hermitian matrices

3/15 Spectral theorem for symmetric matrices (baby version), applications (including graph Laplacian matrix, correlation matrix, Hessian matrix and second derivative test for extrema, unitary matrices, examples including discrete Fourier series and the Fourier matrix, properties

3/20 Spring Break

3/22 Spring Break

3/27 Similarity transformations: examples (including change of basis) and properties, Schur's lemma, spectral theorem for symmetric and Hermitian matrices, unitary matrices are diagonalizable

3/29 Exam 2

4/3 Nilpotent matrices, Jordan canonical form

4/5 JCF, numerical instability of JCF, application to matrix powers (HW: Cayley-Hamilton theorem)

4/10 More examples of Jordan form, positive definite matrices and their properties

4/12 Application of positive/negative definite matrices, additional results about positive definite matrices, statement and proof of Singular Value Decomposition (SVD) theorem, interpretations of SVD, SVD provides orthogonal bases for the four funamental subspace, example

4/17 More applications of SVD: polar decomposition, pseudo-inverses, image compression

4/19 Diagonalizing a covariance matrix via SVD, Principal Component Analysis (PCA) via SVD, Markov chains

4/24 Markov chains (transition matrix, limiting behavior, and ergodicity), umbrella problem, real form of JCF

4/26 Final exam

1/11 How Gauss Elimination works, Gauss Elimination via matrix multiplication, pivots, permutation matrices, PA = LU decomposition, computational cost of LU decomposition, PA=AU via Matlab, inverses

1/16 Matrix inverse via Gauss-Jordan method, linear regression, transpose, symmetric matrices, vector spaces

1/18 Vector spaces and subspaces, matrix groups, 4 fundamental subspaces associated to a matrix, N(A) & C(A) and relations to solving Ax=b

1/23 Echelon form and reduced echelon form, finding N(A), solving Ax=b, numerical solution of boundary value problem

**,**tri-diagonal and band matrices, linear independence and dependence of vectors1/25 More about linear independence, basis, dimensions of N(A) and C(A)

1/30 Fundamental Theorem of Linear Algebra I and corollaries, L/R inverses of mxn matrices

2/1 Numerical issues inverting Hilbert matrix, more about L/R inverses, polynomial interpolation, graphs and incidence matrices, rank-1 matrices, linear transformations, transformations of the plane and computer graphics, vector spaces of polynomials, differentiation via matrix multiplication

2/6 Fundamental Theorem of Calculus for polynomials, inner products, orthogonality, Fundamental Theorem of Orthogonality

2/8 Orthogonal complement, Fundamental Theorem of Linear Algebra II, Fredholm's alternative, orthogonal projection

2/13 Adjoint, Cauchy-Schwarz inequality, orthogonal projection onto a subspace, best approximate solution of Ax=b, projection matrices

2/15 Linear regression revisited, orthonormal bases, orthogonal matrices, Gram-Schmidt orthogonalization

2/20 Exam 1

2/22 Legendre polynomials, Fourier series, A=QR decomposition, application of A=QR to least squares, intersections of subspaces

2/27 Sum of subspaces, direct sum of subspaces, cartesian product of subspaces, isomorphism of vector spaces, tensor product of spaces, definition of determinant via 3 properties

3/1 Properties of determinants, formula for det, existience and unqiueness of det, permutations, det(AB)=det(A)det(B), det of triangular matrix, determinant of any square matrix using PA=PDU, det(A^T)=det(A), determinant via co-factors

3/6 More about co-factors, matrix inverse via co-factors, geometric interpretation of determinants, eigenvalues and eigenvectors, diagonalization

3/8 Solution of linear systems of ODEs x'=Ax, computing matrix powers, Leslie population model, Google's PageRank algorithm

3/13 Perron-Frobenius theorem (statement), square root of a matrix, brief review of complex numbers, complex matrices, Hermitian matrices, spectral properties of symmetric and Hermitian matrices

3/15 Spectral theorem for symmetric matrices (baby version), applications (including graph Laplacian matrix, correlation matrix, Hessian matrix and second derivative test for extrema, unitary matrices, examples including discrete Fourier series and the Fourier matrix, properties

3/20 Spring Break

3/22 Spring Break

3/27 Similarity transformations: examples (including change of basis) and properties, Schur's lemma, spectral theorem for symmetric and Hermitian matrices, unitary matrices are diagonalizable

3/29 Exam 2

4/3 Nilpotent matrices, Jordan canonical form

4/5 JCF, numerical instability of JCF, application to matrix powers (HW: Cayley-Hamilton theorem)

4/10 More examples of Jordan form, positive definite matrices and their properties

4/12 Application of positive/negative definite matrices, additional results about positive definite matrices, statement and proof of Singular Value Decomposition (SVD) theorem, interpretations of SVD, SVD provides orthogonal bases for the four funamental subspace, example

4/17 More applications of SVD: polar decomposition, pseudo-inverses, image compression

4/19 Diagonalizing a covariance matrix via SVD, Principal Component Analysis (PCA) via SVD, Markov chains

4/24 Markov chains (transition matrix, limiting behavior, and ergodicity), umbrella problem, real form of JCF

4/26 Final exam