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Introduction, Vector Spaces, Solutions of Linear Systems, Important Subspaces associated with a matrix, Orthogonality, Eigenvalues and Eigenvectors, Diagonalizable Matrices, Hermitian Matrices, General Matrices, Jordan Canonical form (Optional)*, Selected Topics in Applications (Optional)*

 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 Cayley-Hamilton 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 AAT and ATA Rank, Nullity, Range and Null Space of  AAT and ATA 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 8-10
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