MAT113

UNIT 1

Vector Spaces: Definition, General properties of vector spaces; Vector subspaces; Algebra of subspaces; Linear Spans; Row space of Matrix; Linear dependence and independence of vectors; Finite-dimensional vector spaces; Dimension of vector space and sub-spaces; Quotient spaces; Coordinates.

UNIT 2

Vectors in Rⁿ; Curves in Rⁿ; Matrices: Addition and scalar multiplication, Transpose of matrix, Square matrices; Systems of linear equations; Cayley-Hamilton Theorem; Hermitian & Skew-Hermitian and unitary matrices; Powers of Matrices; Polynomials in Matrices; Invertible Matrices; Special types of Square Matrices; Complex and Block Matrices; Diagonalization; Eigen values and Eigen vectors; Minimal polynomial.

UNIT 3

Linear Transforms; Linear operator; Range and null space of a linear Transformation; Rank and nullity; Product of linear Transformation; Singular Transformation; Representation of linear Transformation by matrix; Dual spaces; Dual Bases.

UNIT 4

Inner Product Spaces: Definition, Euclidean and unitary spaces; Norm and length of vector; Cauchy-Schwarz’s inequality and Applications; Orthogonality, Orthogonal Sets and Basis, Gram- Schmidt orthogonalization process; self-adjoint operators, Complex Inner Product Spaces; Unitary and Normal operators; Spectral theorem.

UNIT 5

Bilinear Forms: Definition, Bilinear form as vectors; Matrix of a bilinear form; Symmetric & skew Symmetric bilinear forms.

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