Vector Spaces: Vector spaces – Sub spaces – Linear independence – Basis – Dimension.
Inner Product Spaces: Inner products – Orthogonality – Orthogonal basis – Gram Schmidt Process – Change of basis – Orthogonal complements – Projection on subspace – Least Square Principle.
Linear Transformations: Positive definite matrices – Matrix norm and condition number – QR- Decomposition – Linear transformation – Relation between matrices and linear transformations – Kernel and range of a linear transformation – Change of basis – Nilpotent transformations – Trace and Transpose, Determinants, Symmetric and Skew Symmetric Matrices, Adjoint and Hermitian Adjoint of a Matrix, Hermitian, Unitary and Normal Transformations, Self Adjoint and Normal Transformations, Real Quadratic Forms.
Eigen values and Eigen vectors: Problems in Eigen Values and Eigen Vectors, Diagonalization, Orthogonal Diagonalization, Quadratic Forms, Diagonalizing Quadratic Forms, Conic Sections. Similarity of linear transformations – Diagonalisation and its applications – Jordan form and rational canonical form.