MSC Semester I -Mathematical Foundations for Data Science (DS)

UNIT- I

Vector Spaces: Vector spaces, Subspaces, Basis and dimension of a vector space, linear

dependence and independence, spanning set. Linear transformation, kemel, range, Matrix

Representation of a linear transformation, rank- nullity theorem, change of basis and similar

matrices Inner-product spaces, orthogonal sets and bases, orthogonal projection, Gram-Schmidt orthogonalization process

UNIT-II

Matrices; trace and Rank of a Matrix and their properties, determinants, inverse, symmetric,

orthogonal and idempotent matrices and their properties, Gauss elimination, row canonical

form, diagonal form, triangular form,

UNIT-III

Eigenvalue Probtems: characteristic roots and vectors, caley-Hamilton theorem,

Diagonalization ofa Matrix, algebraic and geometric multiplicity ofa characteristic root and

spectral decomposition of a rear symmetric matrix., Singular value Decomposition. GaussJordan-LU decomposition, Singular Value Decomposition,

UNIT - IV

Quadratic forms: Real quadratic forms (eFs), reduction and classification ofeFs, index and

signature. Simultaneous reduction of two eFs. Extreme form of a eF. Moore-penrose and

generalized inverses and their properties.