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David Austin

Understanding Linear Algebra

Linear algebra developed visually and computationally: linear systems, matrices and linear combinations, invertibility and bases, eigenvalues, orthogonality and least squares, and the singular value decomposition. The exercises connect geometric intuition to matrix computation and to applications in dynamics and data.

Understanding Linear Algebra

302 exercises32 sections
Status
Sections
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Systems of equations

25
What can we expect2
Finding solutions to linear systems9
Computation with Sage4
Pivots and their influence on solution spaces10

Vectors, matrices, and linear combinations

59
Vectors and linear combinations6
Matrix multiplication and linear combinations15
The span of a set of vectors11
Linear independence12
Matrix transformations11
The geometry of matrix transformations4

Invertibility, bases, and coordinate systems

50
Invertibility15
Bases and coordinate systems9
Image compression1
Determinants14
Subspaces11

Eigenvalues and eigenvectors

46
An introduction to eigenvalues and eigenvectors10
Finding eigenvalues and eigenvectors11
Diagonalization, similarity, and powers of a matrix10
Dynamical systems9
Markov chains and Google's PageRank algorithm6

Linear algebra and computing

21
Gaussian elimination revisited13
Finding eigenvectors numerically8

Orthogonality and Least Squares

53
The dot product9
Orthogonal complements and the matrix transpose14
Orthogonal bases and projections12
Finding orthogonal bases10
Orthogonal least squares8

Singular value decompositions

48
Symmetric matrices and variance12
Quadratic forms11
Principal Component Analysis5
Singular Value Decompositions14
Using Singular Value Decompositions6

Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.