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Vectors and Linear Combinations

Vectors in Rn, scalar multiples, linear combinations, span, the dot product, length and angle, and the column picture of a system. Topics: Vectors and Vector Operations, Linear Combinations and Span, Dot Product, Length, and Angle.

Linear Systems and Elimination

Gaussian elimination, row reduction to RREF, pivots and free variables, the row and column pictures, consistency of Ax=b, and the complete solution. Topics: Geometry of Linear Systems, Gaussian Elimination, Reduced Row Echelon Form, Complete Solution of Ax=b.

Matrix Algebra and Inverses

Matrix arithmetic and multiplication, transpose, the inverse and the invertible-matrix theorem, elimination as matrix multiplication, the A=LU factorization, and permutations. Topics: Matrix Operations, Transpose and Special Matrices, Matrix Inverses, Elimination Matrices and A=LU.

Vector Spaces, Independence, and Basis

Subspaces of Rn, spanning sets, linear independence, basis, dimension, and coordinates, the framework for describing the space a set of vectors generates and measuring its size. Topics: Subspaces of Rn, Linear Independence, Basis, Dimension, and Coordinates, General Vector Spaces (Extension).

The Four Fundamental Subspaces and Rank

Column space, null space, row space, and left null space; rank; the rank-nullity theorem; and the big picture relating Ax=b solvability and Ax=0. Topics: Column Space and Null Space, Row Space and Left Null Space, Rank and the Big Picture.

Linear Transformations

Transformations as matrices, the geometry of common transformations, kernel and image, the matrix of a transformation in chosen bases, and change of basis. Topics: Linear Transformations and Their Matrices, Geometry of Transformations, Kernel and Image, Change of Basis.

Orthogonality and Projection

Orthogonal and orthonormal sets, orthogonal complements, the orthogonality of the four fundamental subspaces, and projection of a vector onto a line or subspace. Topics: Orthogonal and Orthonormal Sets, Orthogonal Complements, Projection onto Subspaces.

Least Squares, Gram-Schmidt, and QR

The normal equations, least-squares fitting and regression, the Gram-Schmidt process, and the A=QR factorization. Topics: Normal Equations and Least Squares, Least-Squares Fitting (Application), Gram-Schmidt and QR.

Determinants

Cofactor expansion, the defining properties, the geometric area/volume meaning, Cramer's rule, and the determinant's role in the characteristic equation. Topics: Computing Determinants, Properties of Determinants, Geometry and Applications of Determinants.

Eigenvalues, Eigenvectors, and Diagonalization

The characteristic equation, eigenspaces, diagonalization, complex eigenvalues, powers of a matrix, and Markov matrices as an application. Topics: Eigenvalues and Eigenvectors, Diagonalization, Complex Eigenvalues, Markov Matrices (Application).

Symmetric Matrices and the Spectral Theorem

Orthogonal diagonalization S = Q Lambda Q-transpose, quadratic forms, and positive-definite matrices. Topics: The Spectral Theorem, Quadratic Forms, Positive-Definite Matrices.

The Singular Value Decomposition

The factorization A = U Sigma V-transpose, its geometry, low-rank approximation, and brief modern applications such as PCA and PageRank. Topics: Computing the SVD, Geometry and Subspaces of the SVD, Low-Rank Approximation and Applications.