Linear Algebra | Xingyu Yang

Reading guide

These notes proceed from linear combinations through matrix operations, systems of equations, vector spaces, linear maps, and graph representations. Inner products and projections add measurable geometry. Determinants, eigenvalues, dynamics, the spectral theorem, and SVD then expose volume, modes, and low-rank structure. Concrete calculations introduce the questions; definitions and proofs explain why the methods work.

Keep three questions separate: whether a target is reachable, whether its representation is unique, and how to compute that representation. Displacements, interpolation, traffic, and webpage transitions appear where they clarify the concepts. Exercises check whether the reasoning transfers to new inputs.

Prerequisites are real arithmetic, elementary equations, and basic sets and functions. Scalars are real unless another field is explicitly specified. The text links to existing chapters when complex numbers, trigonometry, inequalities, recurrences, differential equations, or probability first become relevant.

Reading path

Vectors and Linear Combinations

Build addition, scaling, linear combinations, and reachability from displacement and response models.

Matrices and Composition

Read matrices by rows and columns, model polynomial interpolation, and derive multiplication from composition.

Linear Systems, Elimination, and LU Factorization

Use conservation to understand elimination, free variables, independent measurements, and reusable LU factors.

Vector Spaces, Independence, and Bases

Objects with the same operations, unique coordinates, and the dimension of a solution space.

Linear Maps and Matrix Representations

Represent finite-dimensional linear maps in chosen bases, then connect kernels to solution sets.

Matrix as Graph: Connections, Flows, and Transitions

Distinguish adjacency, incidence, and transition matrices through traffic and webpage models.

Inner Products, Norms, and Orthogonality

Define length and angle through inner products, prove the key inequalities, and construct orthonormal bases.

Orthogonal Projections, Least Squares, and QR

Turn nearest-point problems into orthogonal residuals and derive normal equations, projection matrices, and QR solves.

Determinants and Oriented Volume

Interpret the determinant as oriented volume scaling and derive its behavior under elimination, products, and inversion.

Eigenvalues, Invariant Subspaces, and Diagonalization

Find directions preserved by a transformation and connect eigenspaces, multiplicities, complex eigenvalues, and diagonalization.

Matrix Powers, Recurrences, and Linear Dynamics

Use spectral modes to unify discrete recurrences, Markov updates, and continuous linear differential equations.

Symmetric Matrices, Quadratic Forms, and the Spectral Theorem

Use orthogonal eigenvectors to understand symmetric matrices, quadratic forms, definiteness, and extreme directions.

Singular Value Decomposition, Pseudoinverse, and Low-Rank Approximation

Decompose any rectangular matrix into orthogonal directions and nonnegative scaling, then connect least squares, pseudoinverses, and low-rank models.

Graph Laplacians and Spectral Graph Theory

Construct graph Laplacians from incidence matrices and use eigenvalues to study connectivity, smoothness, diffusion, and clustering.