Carl Meyer

MA 523 - Matrix Analysis and Applied Linear Algebra

Suggested Homework Problems


1. Linear Equations (Review)

1.1 Introduction, Read on your own

1.2 Gaussian Elimination And Matrices, 1---5, 16

1.3 Gauss-Jordan Method, 1---3

1.4 Two-Point Boundary Value Problems, Read on your own

1.5 Making Gaussian Elimination Work, 1---3, 7

1.6 Ill-Conditioned Systems, 1, 2, 4

2. Echelon Forms (Review)

2.1 Row Echelon Form And Rank, 1---4

2.2 The Reduced Row Echelon Form, 1---3

2.3 Consistency Of Linear Systems, 1---7

2.4 Homogeneous Systems, 1---6

2.5 Nonhomogeneous Systems, 1---6

2.6 Electrical Circuits, Read on your own

3. Matrix Algebra (Review)

3.1 From ancient China to Arthur Cayley, Read on your own

3.2 Addition, Scalar Multiplication, And Transposition, Read on your own

3.3 Linearity, 1---2, 4

3.4 Why Do It This Way, Read on your own

3.5 Matrix Multiplication, Read on your own

3.6 Properties Of Matrix Multiplication, Read on your own

3.7 Matrix Inversion, 1---8, 10, 11

3.8 Inverses Of Sums and Sensitivity, 1---3

3.9 Elementary Matrices And Equivalence, 1---6

3.10 The LU Factorization, 1---3, 8, 9, 10

4. Vector Spaces

4.1 Spaces And Subspaces, 1---9

4.2 Four Fundamental Subspaces, 1---6, 9 11, 12, 13

4.3 Linear Independence, 1---7, 9, 10, 11

4.4 Basis And Dimension, 1---6, 8, 9, 16

4.5 More About Rank, 1---8, 9, 16, 20

4.6 Classical Least Squares, 1---4, 6, 9, 10

4.7 Linear Transformations, 1---9, 11---13, 16, 17

4.8 Change Of Basis And Similarity, 1---6, 12

4.9 Invariant Subspaces, 1---4, 8

5. Norms, Inner Products, and Orthogonality

5.1 Vector Norms, 1---6

5.2 Matrix Norms, 1---5

5.3 Inner Product Spaces, 1---5

5.4 Orthogonal Vectors, 1---7, 9, 10, 12---14

5.5 Gram-Schmidt Procedure, 1---6

5.6 Unitary and Orthogonal Matrices, 1---5, 7, 8, 10, 13, 17

5.7 Orthogonal Reduction, 1---4

5.8 Discrete Fourier Transform, 1---6, 8, 19

5.9 Complementary Subspaces, 1---4, 6

5.10 Range-Nullspace Decomposition, 1---6, 10---12

5.11 Orthogonal Decomposition, 1---6, 8, 10, 12

5.12 Singular Value Decomposition, 1, 2, 4, 9, 13, 14, 16, 17

5.13 Orthogonal Projection, 1---7, 9---12

5.14 Why Least Squares? Read on your own

5.15 Angles Between Subspaces, Read on your own

6. Determinants (Review)

6.1 Determinants, Read on your own

6.2 Additional Properties Of Determinants, Read on your own

7. Eigenvalues And Eigenvectors

7.1 Elementary Properties Of Eigensystems, Not yet determined

7.2 Diagonalization by Similarity Transformations, Not yet determined

7.3 Functions Of Diagonalizable Matrices, Not yet determined

7.4 Systems Of Differential Equations, Not yet determined

7.5 Normal Matrices, Not yet determined

7.6 Positive Definite Matrices, Not yet determined

7.7 Nilpotent Matrices And Jordan Structure, Not yet determined

7.8 Jordan Form, Not yet determined

7.9 Functions of Nondiagonalizable Matrices, Not yet determined

7.10 Difference Equations, Limits, Summability, Not yet determined

7.11 Minimum Polynomials and Krylov Methods, Not yet determined