Module No.

Topic/s

Hours

1

Introduction:   

1. First Basic Problem – Systems of Linear equations  - Matrix Notation – The various questions that arise with a system of linear eqautions
2. Second Basic Problem – Diagonalization of a  square matrix – The various questions that arise with diagonalization

3

2

Vector Spaces 

1. Vector spaces
2. Subspaces
3. Linear combinations and subspaces spanned by a set of vectors
4. Linear dependence and Linear independence
5. Spanning Set and Basis
6. Finite dimensional spaces
7. Dimension

6

3

Solutions of Linear Systems

1. Simple systems
2. Homogeneous and Nonhomogeneous systems
3. Gaussian elimination
4. Null Space and Range
5. Rank and nullity
6. Consistency conditions in terms of rank
7. General Solution of a linear system
8. Elementary Row and Column operations
9. Row Reduced Form
10. Triangular Matrix Factorization

6

4

Important Subspaces associsted with a matrix

1. Range and Null space
2. Rank and Nullity
3. Rank Nullity  theorem
4. Four Fundamental subspaces
5. Orientation of the four subspaces

4

5

Orthogonality  

1. Inner product
2. Inner product Spaces
3. Cauchy – Schwarz inequality
4. Norm
5. Orthogonality
6. Gram – Schmidt orthonormalization
7. Orthonormal basis
8. Expansion in terms of orthonormal basis – Fourier series
9. Orthogonal complement
10. Decomposition of a vector with respect to a subspace and its orthogonal complement – Pythagorus Theorem

5

6

Eigenvalues and Eigenvectors

1. What are the ingredients required for diagonalization?
2. Eigenvalue – Eigenvector pairs
3. Where do we look for eigenvalues? – characteristic equation
4. Algebraic multiplicity
5. Eigenvectors, Eigenspaces and geometric multiplicity

5

7

Diagonalizable Matrices

1. Diagonalization criterion
2. The diagonalizing matrix
3. Cayley-Hamilton theorem, Annihilating polynomials,  Minimal Polynomial
4. Diagonalizability and Minimal polynomial
5. Projections
6. Decomposition of the matrix in terms of projections

5

8

Hermitian Matrices

1. Real symmetric and Hermitian Matrices
2. Properties of eigenvalues and eigenvectors
3. Unitary/Orthoginal Diagonalizbility of Complex Hermitian/Real Symmetric matrices
4. Spectral Theorem
5. Positive and Negative Definite and Semi definite matrices

5

9

General Matrices

1. The matrices AA^T and A^TA
2. Rank, Nullity, Range and Null Space of  AA^T and A^TA
3. Strategy for choosing the basis for the four fundamental subspaces
4. Singular Values
5. Singular Value Decomposition
6. Pseudoinverse and Optimal solution of a linear system of equations
7. The Geometry of Pseudoinverse

5

10

Jordan Cnonical form*

1. Primary Decomposition Theorem
2. Nilpotent matrices
3. Canonical form for a nilpotent matrix
4. Jordan Canonical Form
5. Functions of a matrix

5

11

Selected Topics in Applications*

1. Optimization and Linear Programming
2. Network models
3. Game Theory
4. Control Theory
5. Image Compression

8-10