Linear Algebra Study Guide

Systems of Linear Equations and Gaussian Elimination

Master row reduction, echelon forms, and parametric solutions—the computational foundation for all subsequent linear algebra topics.

Matrix Algebra and Matrix Arithmetic

Develop fluency in matrix operations, transpose, and matrix equations essential for representing linear systems compactly.

Matrix Inverses and Elementary Matrices

Compute and apply inverse matrices; understand elementary matrices and LU factorization for solving systems efficiently.

Determinants: Computation and Properties

Calculate determinants via cofactor expansion, apply properties for simplification, and use Cramer's rule and volume interpretation.

Vector Spaces and Subspaces

Understand abstract vector space axioms, identify subspaces, and recognize the unifying structure behind diverse mathematical objects.

Span, Linear Independence, Basis, and Dimension

Master fundamental concepts connecting spanning sets, independence, bases, and dimension for characterizing vector space structure.

Coordinate Systems and Change of Basis

Represent vectors in different bases, compute coordinate vectors, and construct change-of-basis matrices for transformations.

Linear Transformations and Matrix Representations

Define linear maps, construct their matrices, and analyze kernel, range, and the rank-nullity theorem.

Eigenvalues and Eigenvectors

Compute characteristic polynomials, find eigenspaces, understand multiplicities, and apply eigentheory to dynamical systems and Markov chains.

Diagonalization and Matrix Powers

Determine diagonalizability conditions, construct diagonalizing matrices, and efficiently compute matrix powers for applications.

Orthogonality and the Gram-Schmidt Process

Apply inner products, construct orthonormal bases, compute orthogonal projections, and solve least-squares problems with QR factorization.

Symmetric Matrices, Quadratic Forms, and SVD

Apply the spectral theorem, classify quadratic forms, understand positive definiteness, and introduction to singular value decomposition.