Unit 01

Vectors, Linear Systems & Geometric Structure

U01-L01

Vectors and Coordinates in Rⁿ

U01-L02

Dot Products, Norms, Distance & Angle

U01-L03

Linear Combinations and Span

U01-L04

Systems as Intersections and Constraints

U01-L05

The Matrix Equation Ax=b

U01-L06

Consistency and Parametric Solution Sets

U01-L07

Systems Modelling, Residuals & Verification

Unit 02

Matrix Algebra, Elimination & Invertibility

U02-L01

Elementary Row Operations and Echelon Form

U02-L02

Reduced Row Echelon Form and Pivot Structure

U02-L03

Matrix Addition, Multiplication & Composition

U02-L04

Inverse Matrices and Solving Ax=b

U02-L05

The Invertible Matrix Theorem

U02-L06

Elementary Matrices and Row-Operation Algebra

U02-L07

LU Factorization, Repeated Solves & Sensitivity

Unit 03

Matrix Factorizations & Computational Structure

U03-L01

Block Matrices and Structured Products

U03-L02

Permutation Matrices and Pivoting

U03-L03

Computational Cost and Matrix Algorithms

U03-L04

LU with Partial Pivoting

U03-L05

QR Factorization Preview

U03-L06

Choosing a Factorization

U03-L07

Norms, Backward Error & Verification

Unit 04

Vector Spaces, Subspaces & Generated Structure

U04-L01

Vector-Space Axioms and Examples

U04-L02

The Subspace Test

U04-L03

Column Space and Solvability

U04-L04

Nullspace and Homogeneous Solutions

U04-L05

Row Space, Left Nullspace & Orthogonality

U04-L06

Polynomial, Function & Matrix Spaces

U04-L07

Sums, Intersections & the Four Subspaces

Unit 05

Linear Independence, Basis, Dimension & Rank

U05-L01

Linear Independence and Dependence Relations

U05-L02

Spanning Sets and Basis Extraction

U05-L03

Coordinate Vectors Relative to a Basis

U05-L04

Dimension and Extension/Reduction Theorems

U05-L05

The Rank–Nullity Theorem

U05-L06

Change-of-Basis Matrices

U05-L07

Basis Construction Algorithms & Proof

Unit 06

Linear Transformations & Matrix Representations

U06-L01

Definition and Tests for Linearity

U06-L02

Kernel, Image & Rank–Nullity for Transformations

U06-L03

The Standard Matrix of a Transformation

U06-L04

Composition and Inverse Transformations

U06-L05

Geometric Transformations in the Plane and Space

U06-L06

Matrix Representations Relative to Bases

U06-L07

Isomorphisms, Similarity & Structural Equivalence

Unit 07

Determinants, Orientation & Volume

U07-L01

Determinant Properties and Transformations

U07-L02

Cofactor Expansion and Recursive Structure

U07-L03

Determinants by Elimination

U07-L04

Signed Area, Volume & Orientation

U07-L05

Cramer’s Rule and Theoretical Solves

U07-L06

Determinant, Invertibility & Eigenvalue Zero

U07-L07

Multilinearity, Alternation & Proof

Unit 08

Eigenvalues, Eigenvectors & Diagonalization

U08-L01

Invariant Directions and Eigenpairs

U08-L02

Characteristic Polynomial and Spectrum

U08-L03

Eigenspaces and Multiplicity

U08-L04

Diagonalization Criteria and Construction

U08-L05

Matrix Powers, Difference Equations & Long-Term Behavior

U08-L06

Complex Eigenvalues, Rotations & Real Dynamics

U08-L07

Defective Matrices and Jordan-Form Preview

Unit 09

Inner Products, Orthogonality & Least Squares

U09-L01

Inner Products and Orthogonality

U09-L02

Orthogonal Complements and Decomposition

U09-L03

Projection onto a Vector or Subspace

U09-L04

Least Squares and the Normal Equations

U09-L05

Gram–Schmidt Orthonormalization

U09-L06

Orthogonal Matrices and QR Factorization

U09-L07

Linear Regression, Residual Diagnostics & Model Limits

Unit 10

Symmetric Matrices, Quadratic Forms & the Spectral Theorem

U10-L01

Symmetric Matrices and Orthogonal Eigenvectors

U10-L02

The Spectral Theorem

U10-L03

Positive Definite Matrices and Tests

U10-L04

Quadratic Forms and Classification

U10-L05

Cholesky Factorization

U10-L06

Rayleigh Quotients and Constrained Optimization

U10-L07

Principal Axes, Covariance & Geometric Interpretation

Unit 11

Singular Value Decomposition & Data Geometry

U11-L01

Constructing the Singular Value Decomposition

U11-L02

Geometry of the SVD

U11-L03

The Moore–Penrose Pseudoinverse

U11-L04

Low-Rank Approximation and Eckart–Young

U11-L05

Principal Component Analysis

U11-L06

Image Compression and Reconstruction

U11-L07

Conditioning, Noise & Truncated SVD

Unit 12

Applications: Markov Models, Networks, Graphics & Coding

U12-L01

Markov Chains and Stationary Distributions

U12-L02

Graphs, Incidence Matrices & Laplacians

U12-L03

Network Flows and Conservation

U12-L04

Computer Graphics and Homogeneous Coordinates

U12-L05

Error-Correcting Codes over Finite Fields

U12-L06

PageRank, Ranking & Recommendation Models

U12-L07

Coupled Differential Systems and Matrix Exponentials

Unit 13

Numerical Linear Algebra, Proof & Capstone

U13-L01

Floating-Point Arithmetic and Error Propagation

U13-L02

Iterative Methods for Linear Systems

U13-L03

Sparse Matrices and Large-Scale Structure

U13-L04

Proof Strategies in Linear Algebra

U13-L05

Capstone Question, Data, Assumptions & Method

U13-L06

Capstone Computation, Verification & Reproducibility

U13-L07

Written Report and Oral Mathematical Defense

U13-L08

Second-Course Bridge: Abstract, Numerical & Applied Directions