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Systems of Linear Equations
Row reduction, Gaussian elimination, and solution types.
Vectors and Euclidean Spaces
Vector arithmetic, linear combinations, span, and independence.
Matrix Algebra
Matrix operations, inverses, and characterizations of invertibility.
Determinants
Cofactor expansion, properties, Cramer's rule, and geometric interpretation.
Vector Spaces
Subspaces, basis, dimension, rank, and the rank-nullity theorem.
Orthogonality and Least Squares
Orthogonal projections, Gram-Schmidt process, and least-squares fitting.
Eigenvalues and Eigenvectors
Characteristic equation, diagonalization, and complex eigenvalues.
Symmetric Matrices and Quadratic Forms
Spectral theorem, quadratic forms, SVD, and PCA.