Matrices

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.