Week 1:Introduction to matrices, matrix operations, such as addition of matrices, transpose of a matrix, scalar multiplication and matrix multiplication etc. Invertible matrices, examples and submatrix of a matrix
Week 2:System of linear equations, elementary row operations, elementary matrices, Gauss elimination and Gauss – Jordan methods, LU decomposition, Row-reduced echelon form of a matrix, Rank of a matrix and the solution set of a linear system
Week 3: Application of the solution set of a linear system to linear systems where the coefficient matrix is a square matrix. Determinant of a matrix, Inverse using the classical adjoint method and the Cramer’s rule.
Week 4:Axiomatic definition of a vector space, examples, subspaces and linear combination and linear span, finite dimensional vector space, fundamental subspaces associated with a matrix
Week 5:Linear independence and dependence, linear independence and the rank of a matrix, basis of a vector space, constructing a basis of a finite dimensional vector space
Week 6:Linear transformations, rank-nullity theorem and its application to maps between finite dimensional vector spaces.
Week 7: Ordered bases, matrix of a linear transformation and similarity of matrices
Week 8:Dot/Inner product in a vector space, Cauchy Schwartz inequality, angle between two vectors, Projection of a vector onto another vector
Week 9:Gram-Schmidt orthogonalization process and the QR- decomposition, least square solution of a non-consistent linear system and the orthogonal projections,
Week 10:Motivation, definition and examples for eigenvalues and eigenvectors, Schur’s unitary triangularization,
Week 11: Diagonalizable matrices, Criteria for diagonalizability, diagonalizability of Normal matrices, Spectral theorem for Hermitian matrices
Week 12:Quadratic forms, Sylvester’s law of inertia, Classification of quadrics
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