AI Tutor
Back to library

Jim Hefferon

Linear Algebra

A proof-based first course in linear algebra: linear systems and Gauss’s Method, vector spaces, maps between spaces, determinants, and similarity and eigenvalues. The exercises range from computation to proof, with fully worked answers that model how to write mathematics well.

Linear Algebra

959 exercises63 sections
Status
Sections
FlagsLoading your progress…

Linear Systems

130
Gauss’s Method25
Describing the Solution Set20
General=Particular+Homogeneous12
Vectors in Space10
Length and Angle Measures28
Gauss-Jordan Reduction14
The Linear Combination Lemma17
Accuracy of Computations4

Vector Spaces

200
Definition and Examples29
Subspaces and Spanning Sets30
Definition and Examples26
Basis23
Dimension26
Vector Spaces and Linear Systems28
Combining Subspaces21
Fields5
Crystals3
Voting Paradoxes3
Dimensional Analysis6

Maps Between Spaces

339
Definition and Examples27
Dimension Characterizes Isomorphism20
Definition27
Range Space and Null Space26
Representing Linear Maps with Matrices21
Any Matrix Represents a Linear Map22
Sums and Scalar Products11
Matrix Multiplication23
Mechanics of Matrix Multiplication27
Inverses26
Changing Representations of Vectors17
Changing Map Representations22
Orthogonal Projection Into a Line15
Gram-Schmidt Orthogonalization18
Projection Into a Subspace19
Line of Best Fit1
Geometry of Linear Maps5
Magic Squares5
Markov Chains2
Orthonormal Matrices5

Determinants

138
Exploration19
Properties of Determinants18
The Permutation Expansion25
Determinants Exist9
Determinants as Size Functions20
Laplace’s Expansion19
Cramer’s Rule8
Speed of Calculating Determinants5
Chiò’s Method4
Projective Geometry6
Computer Graphics5

Similarity

152
Definition and Examples20
Diagonalizability15
Eigenvalues and Eigenvectors27
Self-Composition8
Strings17
Polynomials of Maps and Matrices22
Jordan Canonical Form24
Method of Powers2
Stable Populations1
Page Ranking1
Linear Recurrences5
Coupled Oscillators4
Inner Product6

Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.