Foundation

Vector Spaces, Subspaces, Basis, Linear transformation, Quotient Space, Rank-nullity theorem, Matrix representations, Inner product Spaces

Diagonalization

Eigenvalues and Eigenvectors; Geometric and Algebraic multiplicities, characteristic equation, spectral theorem for a symmetric operator in a f.d. inner
product space. Structure of a single linear transformation, Algebra of Projections, Equivalence of Matrices and Similarity of Matrices

Isomorphism Theorems

Isomorphism theorems, Linear Functionals, Dual bases, Operator Adjoints, norm, orthogonal direct sums,The Riesz Representation Theorem

Algebraic Structures

Groups, Subgroups, Ring, Ideals, Quotient Rings, Maximal Ideals, Quotient fields, PIDs, UIDs, Fields and Algebras

Module

Modules, Submodules, Homomorphisms , Quotient Modules, The Cyclic Decomposition of a Primary Module