Module No. 
Topic/s 
Hours 
1 
Introduction:
 First Basic Problem – Systems of Linear equations  Matrix Notation – The various questions that arise with a system of linear eqautions
 Second Basic Problem – Diagonalization of a square matrix – The various questions that arise with diagonalization

3 
2 
Vector Spaces
 Vector spaces
 Subspaces
 Linear combinations and subspaces spanned by a set of vectors
 Linear dependence and Linear independence
 Spanning Set and Basis
 Finite dimensional spaces
 Dimension

6 
3 
Solutions of Linear Systems
 Simple systems
 Homogeneous and Nonhomogeneous systems
 Gaussian elimination
 Null Space and Range
 Rank and nullity
 Consistency conditions in terms of rank
 General Solution of a linear system
 Elementary Row and Column operations
 Row Reduced Form
 Triangular Matrix Factorization

6 
4 
Important Subspaces associsted with a matrix
 Range and Null space
 Rank and Nullity
 Rank Nullity theorem
 Four Fundamental subspaces
 Orientation of the four subspaces

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

5 
6 
Eigenvalues and Eigenvectors
 What are the ingredients required for diagonalization?
 Eigenvalue – Eigenvector pairs
 Where do we look for eigenvalues? – characteristic equation
 Algebraic multiplicity
 Eigenvectors, Eigenspaces and geometric multiplicity

5 
7 
Diagonalizable Matrices
 Diagonalization criterion
 The diagonalizing matrix
 CayleyHamilton theorem, Annihilating polynomials, Minimal Polynomial
 Diagonalizability and Minimal polynomial
 Projections
 Decomposition of the matrix in terms of projections

5 
8 
Hermitian Matrices
 Real symmetric and Hermitian Matrices
 Properties of eigenvalues and eigenvectors
 Unitary/Orthoginal Diagonalizbility of Complex Hermitian/Real Symmetric matrices
 Spectral Theorem
 Positive and Negative Definite and Semi definite matrices

5 
9 
General Matrices
 The matrices AA^{T} and A^{TA}
 Rank, Nullity, Range and Null Space of AA^{T} and A^{T}A
 Strategy for choosing the basis for the four fundamental subspaces
 Singular Values
 Singular Value Decomposition
 Pseudoinverse and Optimal solution of a linear system of equations
 The Geometry of Pseudoinverse

5 
10 
Jordan Cnonical form*
 Primary Decomposition Theorem
 Nilpotent matrices
 Canonical form for a nilpotent matrix
 Jordan Canonical Form
 Functions of a matrix

5 
11 
Selected Topics in Applications*
 Optimization and Linear Programming
 Network models
 Game Theory
 Control Theory
 Image Compression

810 