MA 405

COURSE OUTLINE


1. Linear Equations (Review)

1.1 Introduction

1.2 Gaussian Elimination And Matrices

1.3 Gauss-Jordan Method

1.5 Making Gaussian Elimination Work

1.6 Ill-Conditioned Systems

2. Echelon Forms (Review)

2.1 Row Echelon Form And Rank

2.2 The Reduced Row Echelon Form

2.3 Consistency Of Linear Systems

2.4 Homogeneous Systems

2.5 Nonhomogeneous Systems

3. Matrix Algebra (Review)

3.2 Addition, Scalar Multiplication, And Transposition

3.3 Linearity

3.5 Matrix Multiplication

3.6 Properties Of Matrix Multiplication

3.7 Matrix Inversion

3.8 Inverses Of Sums and Sensitivity

3.9 Elementary Matrices And Equivalence

3.10 The LU Factorization

4. Vector Spaces

4.1 Spaces And Subspaces

4.2 Four Fundamental Subspaces

4.3 Linear Independence

4.4 Basis And Dimension

4.5 More About Rank

4.6 Classical Least Squares

4.7 Linear Transformations

4.8 Change Of Basis And Similarity

4.9 Invariant Subspaces

5. Norms, Inner Products, and Orthogonality

5.1 Vector Norms And Inner Products

5.2 Orthogonal Vectors

5.3 Gram-Schmidt Procedure

5.4 Unitary and Orthogonal Matrices

5.5 Orthogonal Reduction

5.7 Complementary Subspaces

5.8 Range-Nullspace Decomposition

5.9 Orthogonal Decomposition

5.10 Singular Value Decomposition

5.11 Orthogonal Projection

5.12 Why Least Squares?

5.13 Angles Between Subspaces

6. Determinants (Review)

6.1 Determinants

6.2 Additional Properties Of Determinants

7. Eigenvalues And Eigenvectors

7.1 Elementary Properties Of Eigensystems

7.2 Diagonalization by Similarity Transformations

7.3 Functions Of Diagonalizable Matrices

7.4 Systems Of Differential Equations

7.5 Normal Matrices

7.6 Positive Definite Matrices

7.7 Non-Diagonalizale Matrices And Jordan Form