1. 1. Systems of Linear Equations and Matrices (8 Lectures) 1.1 Introduction to Systems of Linear Equations 1.2 Gaussian Elimination 1.3 Matrices and Matrix Operations 1.4 Inverses: Algebraic Properties of Matrices 1.5 Elementary Matrices and a method to find A -1 1.6 More on Linear Systems and Invertible Matrices 1.7 Diagonal, Tridiagonal, and Symmetric Matrices 1.8 Applications: Leontief Input-Output Models

2. 2. Determinants (3 Lectures) 2.1 Determinants by Cofactor Expansion 2.2 Evaluating Determinants by Row Reduction 2.3 Properties of Determinants: Cramer’s Rule Definition of Determinant, det(A) or |A| Cofactor and minor at (i,j)-position of A Properties of determinants Examples and Applications

3. 3.+4. Vector Space (4+10 Lectures) 8. Linear Transform (4 Lectures) Vector Space, Subspace, Examples Null space, column space, row space of a matrix Spanning sets, Linear Independence, Basis, Dimension Rank, Nullity, and the Fundamental Matrix Spaces Matrix Transformation from R n to R m Kernel and Image of a Linear Transform Projection, rotation, scaling Gauss transform Householder transform (elementary reflector) Jacobi transform (Givens’ rotation) Isomorphism for L: V  W

4. 5. Eigenvalues and Eigenvectors (3 Lectures) 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 5.4 Differential Equations

5. 6. Inner Product Spaces (3 Lectures) 6.1 Inner Products 6.2 Angle and Orthogonality in Inner Product Spaces 6.3 Gram-Schmidt Process: QR-Decomposition 6.4 Best Approximation: Least Squares 6.5 Least Square Fitting to Data 6.6 Function Approximation: Fourier Series

6. 7. Diagonalization and Quadratic Forms (4 Lectures) 7.1 Orthogonality Matrices 7.2 Orthogonal Diagonalization 7.3 Quadratic Forms 7.4 Optimization Using Quadratic Forms 7.5 Hermitian, Unitary, and Normal Matrices

7. 9. Numerical Methods (Optional) 9.1 LU-Decomposition 9.2 The Power Method 9.3 Internet Search Engine 9.4 Comparison of Procedures for Solving Linear Systems of Equations 9.5 Singular Value Decomposition (SVD) 9.6 Data Compression Using Singular Value Decomposition

8. 10. Applications of Linear Algebra (Optional) 10.1 Constructing Curves and Surfaces Through Specified Points 10.4. Cubic Spline Interpolation 10.6 Graph Theory 10.7 Game Theory 10.8 Leontief Economic Models 10.10 Computer Graphics 10.11 Equilibrium Temperature Distribution 10.13 Fractals 10.15 Cryptography 10.16 Warps and Morphs