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