Contents:
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

Other Resources

Course Curriculum

Prologue Part 1 Details 57:44
Prologue Part 2 Details 52:25
Prologue Part 3 Details 53:5
Linear Systems Part 1 Details 53:41
Linear Systems Part 2 Details 55:24
Linear Systems Part 3 Details 54:51
Linear Systems Part 4 Details 55:42
Vector Spaces Part 1 Details 54:52
Vector Spaces Part 2 Details 55:16
Linear Independence and Subspaces Part 1 Details 56:51
Linear Independence and Subspaces Part 2 Details 56:6
Linear Independence and Subspaces Part 3 Details 59:21
Linear Independence and Subspaces Part 4 Details 55:24
Basis Part 1 Details 58:23
Basis Part 2 Details 56:12
Basis Part 3 Details 57:17
Linear Transformations Part 1 Details 56:54
Linear Transformations Part 2 Details 56:53
Linear Transformations Part 3 Details 56:52
Linear Transformations Part 4 Details 59:3
Linear Transformations Part 5 Details 58:1
Inner Product and Orthogonality Part 1 Details 55:11
Inner Product and Orthogonality Part 2 Details 54:7
Inner Product and Orthogonality Part 3 Details 55:59
Inner Product and Orthogonality Part 4 Details 56:25
Inner Product and Orthogonality Part 5 Details 56:29
Inner Product and Orthogonality Part 6 Details 52:43
Diagonalization Part 1 Details 55:34
Diagonalization Part 2 Details 57:11
Diagonalization Part 3 Details 57:23
Diagonalization Part 4 Details 59:40
Hermitian and Symmetric matrices Part 1 Details 54:37
Hermitian and Symmetric matrices Part 2 Details 56:18
Hermitian and Symmetric matrices Part 3 Details 57:40
Hermitian and Symmetric matrices Part 4 Details 57:6
Singular Value Decomposition (SVD) Part 1 Details 57:52
Singular Value Decomposition (SVD) Part 2 Details 57:19
Back To Linear Systems Part 1 Details 57:37
Back To Linear Systems Part 2 Details 58:32
Epilogue Details 56:13

This Course is provided by IISc Bangalore as part of NPTEL online courses.

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