**Lecture Schedule**

1/9 Introduction, Google's PageRank, upper/lower trangular (UT/LT) matrices, solving Ax=b for an UT matrix

1/11 Gaussian Elimination by elementary row operations and by elementary matrices

1/13 A=LU and LDU decompositions, application to solving Ax=b, computational cost O(n^3)

1/16 MLK day (no class)

1/18 Class cancelled

1/20 Properties of matrix inverse, Gauss-Jordan algorithm

1/23 Properties of matrix transpose, simple linear regression, numerically solving 2nd order ODE BVP, tri-diagonal and positive definite matrices

1/25 Vector spaces and subspaces, col(A)

1/27 Row echelon form of matrix, N(A)

1/30 Solving Ax=b, affine subspaces, linear dependence and independence

2/1 More on linear dependence and independence, basis, dimension

2/3 The four fundamental spaces for a matrix, Fundamental Theorem of Linear Algebra I, rank theorem

2/6 Existence and uniqueness of solutions of Ax=b, left and right inverses, linear transformations

2/8 Differentiation and integration via matrices and FTC, every linear transformation is given by matrix multiplication, inner products, length of a vector, orthogonal vectors

2/10 Polynomial interpolation, orthogonal subspaces, orthogonal complement, Fundamental Theorem of Linear Algebra II

2/13 Proof of FTLA II and applications, Strang's favorite figure, orthogonal projection, projection matrices, adjoints

2/15

2/17 Orthogonal projection onto a subspace, least squares, best approx solution of Ax=b when b is not in col(A)

2/20 Formulas for best approx solution and orthogonal projection of b onto col(A), projection matrices, orthogonal bases and orthogonal matrices

2/22 More on orthogonality, Fredholm alternative, Gram-Schmidt orthogonalization method

2/24 Examples of Gram-Schmidt method, Legendre polynomials, A=QR factorization

2/27 Intersection, sum, and direct sum of subspaces

3/1 Products of subspaces, isomorphism of vector spaces, tensor product of subspaces, defining properties of det(A)

3/3 Properties of det(A)

3/6 Permutations, determinant via co-factors, geometric interpretations of determinants: volume of parallelepipeds and distortion of volume under linear mappings

3/8 Eigenvalues and eigenvectors, eigenvalues of special matrices, Fundamental Theorem of Algebra

3/10 Diagonalization, solving linear systems of ODEs

3/13 ODEs continued, matrix powers, Leslie population model

3/15

3/17 Linear difference equations, e^A, solving x'=Ax as x(t)=e^{tA}c, e^z, e^{i x}=i sin(x)+ cos(x), e^{i pi}=-1

3/20 Spring Break

3/22 Spring Break

3/24 Spring Break

3/27 Complex matrices, Hermitian matrices, spectral properties of symmetric and Hermitian matrices

3/29 Spectral theorem for symmetric matrices

3/31 Unitary matrices

4/3 Spectral theorems for Hermitian and unitary matrices

4/5 Class cancelled

4/7 Guest lecture by Professor Bellissard on quantum computing

4/10 Similarity and Jordan Canonical Form

4/12 Jordan Canonical Form

4/14 Jordan Canonical Form

4/17 Applications of Jordan Canonical Form

4/19 Postive definite matrices

4/21 Singular value decomposition

4/24 Applications of SVD to image compression and principal component analysis

4/28

1/11 Gaussian Elimination by elementary row operations and by elementary matrices

1/13 A=LU and LDU decompositions, application to solving Ax=b, computational cost O(n^3)

1/16 MLK day (no class)

1/18 Class cancelled

1/20 Properties of matrix inverse, Gauss-Jordan algorithm

1/23 Properties of matrix transpose, simple linear regression, numerically solving 2nd order ODE BVP, tri-diagonal and positive definite matrices

1/25 Vector spaces and subspaces, col(A)

1/27 Row echelon form of matrix, N(A)

1/30 Solving Ax=b, affine subspaces, linear dependence and independence

2/1 More on linear dependence and independence, basis, dimension

2/3 The four fundamental spaces for a matrix, Fundamental Theorem of Linear Algebra I, rank theorem

2/6 Existence and uniqueness of solutions of Ax=b, left and right inverses, linear transformations

2/8 Differentiation and integration via matrices and FTC, every linear transformation is given by matrix multiplication, inner products, length of a vector, orthogonal vectors

2/10 Polynomial interpolation, orthogonal subspaces, orthogonal complement, Fundamental Theorem of Linear Algebra II

2/13 Proof of FTLA II and applications, Strang's favorite figure, orthogonal projection, projection matrices, adjoints

2/15

**Exam 1**2/17 Orthogonal projection onto a subspace, least squares, best approx solution of Ax=b when b is not in col(A)

2/20 Formulas for best approx solution and orthogonal projection of b onto col(A), projection matrices, orthogonal bases and orthogonal matrices

2/22 More on orthogonality, Fredholm alternative, Gram-Schmidt orthogonalization method

2/24 Examples of Gram-Schmidt method, Legendre polynomials, A=QR factorization

2/27 Intersection, sum, and direct sum of subspaces

3/1 Products of subspaces, isomorphism of vector spaces, tensor product of subspaces, defining properties of det(A)

3/3 Properties of det(A)

3/6 Permutations, determinant via co-factors, geometric interpretations of determinants: volume of parallelepipeds and distortion of volume under linear mappings

3/8 Eigenvalues and eigenvectors, eigenvalues of special matrices, Fundamental Theorem of Algebra

3/10 Diagonalization, solving linear systems of ODEs

3/13 ODEs continued, matrix powers, Leslie population model

3/15

**Exam 2**3/17 Linear difference equations, e^A, solving x'=Ax as x(t)=e^{tA}c, e^z, e^{i x}=i sin(x)+ cos(x), e^{i pi}=-1

3/20 Spring Break

3/22 Spring Break

3/24 Spring Break

3/27 Complex matrices, Hermitian matrices, spectral properties of symmetric and Hermitian matrices

3/29 Spectral theorem for symmetric matrices

3/31 Unitary matrices

4/3 Spectral theorems for Hermitian and unitary matrices

4/5 Class cancelled

4/7 Guest lecture by Professor Bellissard on quantum computing

4/10 Similarity and Jordan Canonical Form

4/12 Jordan Canonical Form

4/14 Jordan Canonical Form

4/17 Applications of Jordan Canonical Form

4/19 Postive definite matrices

4/21 Singular value decomposition

4/24 Applications of SVD to image compression and principal component analysis

4/28

**Final Exam**