Matrices and Linear Algebra

Matrix Inversion: The Adjugate Matrix and Invertibility Criteria

Determinants: Axiomatic Definition, Laplace Expansion and Properties

Fundamental Subspaces: Row Space, Column Space and Null Space

Euclidean Spaces and Inner Products

Systems of Linear Equations: Gaussian Elimination and Gauss-Jordan Reduction

Generalized Eigenspaces and the Jordan Canonical Form

Bilinear and Quadratic Forms: Matrix Congruence and Sylvester’s Criterion

Matrix Representation of Linear Operators and Coordinate Transformations

Eigenvalues, Eigenvectors and the Characteristic Polynomial

Systems of Linear Equations: Cramer’s Rule and the Rouché-Capelli Theorem

Vector Spaces, Subspaces, Span, Basis and Dimension

Matrix Similarity and Diagonalization Criteria

Matrix Operations, Algebraic Properties and Special Matrix Classes

Elementary Row Operations, Echelon Forms and Matrix Equivalence

Linear Transformations: Kernel and Image

Matrix Factorizations: LU Decomposition, QR Factorization and SVD

Matrix Rank, Nullity and the Rank-Nullity Theorem

The Cayley-Hamilton Theorem and Minimal Polynomials