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.
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
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
Important Subspaces associsted with a matrix : Range and Null space – Rank and Nullity – Rank Nullity theorem – Four Fundamental subspaces – Orientation of the four subspaces
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
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
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
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
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
Jordan Cnonical form* : Primary Decomposition Theorem – Nilpotent matrices – Canonical form for a nilpotent matrix – Jordan Canonical Form – Functions of a matrix.
Selected Topics in Applications* : Optimization and Linear Programming – Network models – Game Theory – Control Theory – Image Compression
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