LINEAR ALGEBRA

Linear transformation of vector spaces, Dual spaces, Dual basis and their
properties, Dual maps, Annihilator.

Matrices of a linear maps, Matrices of composition maps, Matrices of dual map, Eigen values, Eigen vectors, Rank and Nullity of linear maps and matrices, Invertible matrices, Similar matrices, Diagonalization of Matrices.

Determinants of matrices and its computations, Characteristic polynomial and Eigen values. Minimal polynomial, Cayley-Hamiltton theorem.

Bilinear forms: Definition and examples. The matrix of a Bilinear form, Orthogonality, Classification of Bilinear forms.
 

Real inner product space, Schwartz’s inequality, Orthogonality, Bessel’s inequality, Adjoint, Self adjoint linear transformations and matrices, orthogonal linear transformation and matrices, Principal Axis Theorem.