SYLLABUS- MATRICES, Section-A: Linear Algebra Unit-1: Vector spaces and their elementary properties, Subspaces, Linear dependence and independence, Basis and dimension, Direct sum, Quotient space. Unit-2: Linear transformations and their algebra, Range and null space, Rank and nullity, Matrix representation of linear transformations, Change of basis. Unit-3: Linear functionals, Dual space, Bi-dual space, Natural isomorphism, Annihil- ators, Bilinear and quadratic forms, Inner product spaces, Cauchy-Schwarz's inequality, Bessel's inequality and orthogonality. Section-B: Matrices Unit-4: Symmetric and skew-symmetric matrices, Hermitian and skew-Hermitian matrices, Orthogonal and unitary matrices, Triangular and diagonal matrices, Rank of a matrix, Elementary transformations, Echelon and normal forms, Inverse of a matrix by elementary transformations. Unit-5: Characteristic equation, Eigen values and eigen vectors of a matrix, Cayley- Hamilton's theorem and its use in finding inverse of a matrix, Application of matrices to solve a system of linear (both homogeneous and non-homogeneous) equations, Consistency and general solution, Diagonalization of square matrices with distinct eigen values, Quadratic forms.

Krishna's TB Matrices PDF

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