4 Topics Available

Linear Algebra

Vector spaces, matrix transformations, Gaussian elimination, eigenvalues, eigenvectors, and Singular Value Decomposition (SVD).

1Vector Spaces, Linear Independence & Subspace Bases

Span, linear combinations, basis vectors, dimension, and fundamental subspaces of a matrix.

Vector Subspaces, Linear Independence & Span Mechanics

Mathematical definitions of vector space axioms, linear combinations, and basis dimensions.

Vector Space Axioms, Linear Independence & Basis Dimension

Formal vector space axioms, linear combination span, linear independence testing via determinant, and rank-nullity theorem.

2Inner Product Spaces, Orthogonality & QR Decomposition

Inner product spaces, Cauchy-Schwarz bounds, Gram-Schmidt orthogonalization, Householder reflections, and QR matrix factorization.

Orthogonal Subspaces, Projection Geometry & QR Factorization

Vector norms, projection matrices, Modified Gram-Schmidt stability, and QR factorizations for overdetermined linear systems.

Inner Product Spaces, Modified Gram-Schmidt & QR Matrix Factorization

Examine inner product axioms, orthogonal projections onto linear subspaces, numerical instability in classical Gram-Schmidt, and the implementation of Modified Gram-Schmidt QR decomposition.

3Eigendecomposition, Diagonalization & Spectral Theory

Characteristic polynomials, eigenvalues, eigenvectors, algebraic vs geometric multiplicity, and the Spectral Theorem for symmetric matrices.

Eigenvalues, Characteristic Polynomials & Diagonalization

Invariant subspaces, eigenvalue computation, diagonalizability criteria, and real symmetric matrix spectral decompositions.

Characteristic Polynomials, Eigendecomposition & the Spectral Theorem

Master invariant vector directions, the characteristic determinant equation, geometric versus algebraic multiplicities, matrix diagonalizability criteria, and the Spectral Theorem for real symmetric operators.

4Singular Value Decomposition (SVD) & Low-Rank Approximation

Singular values, singular vectors, Eckart-Young-Mirsky Theorem, Moore-Penrose pseudoinverse, and data dimensionality reduction.

SVD Geometry, Fundamental Subspaces & Low-Rank Factorization

Left/right singular vectors, singular value spectrum, low-rank matrix compression, and robust pseudo-inversion for rank-deficient systems.

Singular Value Decomposition, Moore-Penrose Pseudoinverse & Low-Rank Approximation

Master the geometry of SVD, the connection between singular values and the four fundamental subspaces, the Eckart-Young-Mirsky low-rank theorem, Moore-Penrose pseudoinverses, and production image compression.