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MATH 247 Applied Linear Algebra - Coggle Diagram
MATH 247 Applied Linear Algebra
Preliminary
Fields, Subfields
The field of complex numbers
Matrices over a field
Vector space
Subspaces
Linear hull / span, independence
Definition / properties
Basis / Dimension
Bound Lemma
Basis Selection
Basis Completion
Finite dimensional vector spaces
Basis theorems
Dimension theorems for subspaces
Direct sums
Linear Maps
Definition and elementary properties
Linear Extension Theorem
kernel-defect, range-rank
Properties
Injective/Surjective/Bijective
Rank-nullity Theorem
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
Matrices revisited
Matrix representation of linear maps
Rank equivalence of matrices
Matrices as linear mappings
Equivalence of matrices and rank normal form
Groups
Elementary operations on matrices
The determinant of a square matrix
Eigenvalue Theory
Diagonalizability I
The characteristic polynomial
Real and complex polynomials
The characteristic polynomial of matrices and endomorphisms
Eigenvalues and eigenvectors
Similar matrices
Diagonalizability II
The minimal polynomial
Triangularizability
Inner Product Spaces
Classes of normal endomorphisms
The singular value decomposition
Adjoint map and normal endomorphisms
Angles and orthogonality
Inner products and norms