MATH 247 Applied Linear Algebra

Preliminary

Vector space

Fields, Subfields

The field of complex numbers

Matrices over a field

Linear Maps

Matrices revisited

Eigenvalue Theory

Subspaces

Linear hull / span, independence

Definition / properties

Basis / Dimension

Finite dimensional vector spaces

Direct sums

Basis theorems

Dimension theorems for subspaces

Definition and elementary properties

Rank-nullity Theorem

Matrix representation of linear maps

Rank equivalence of matrices

Groups

Elementary operations on matrices

The determinant of a square matrix

Matrices as linear mappings

Equivalence of matrices and rank normal form

Diagonalizability I

The characteristic polynomial

Eigenvalues and eigenvectors

Similar matrices

Real and complex polynomials

The characteristic polynomial of matrices and endomorphisms

Inner Product Spaces

Classes of normal endomorphisms

The singular value decomposition

Adjoint map and normal endomorphisms

Angles and orthogonality

Inner products and norms

Diagonalizability II

The minimal polynomial

Triangularizability

Bound Lemma

Basis Selection

Basis Completion

Linear Extension Theorem

kernel-defect, range-rank

Properties

Injective/Surjective/Bijective

Rank inequalities

def(f)+rank(f) = dim(f)

isomorphism theorem

Every n-dimensional v.s. over K is isomorphic to K^n

Two n-dimensional v.s are isomorphic