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Peter Selinger

Matrix Theory and Linear Algebra

A full first course in linear algebra grown from the Kuttler/Lyryx text: systems of equations, vectors, matrices, determinants, eigenvalues, abstract vector spaces, linear transformations, and inner product spaces, with applications from resistor networks to principal component analysis. Most exercises come with the book's own answers.

Matrix Theory and Linear Algebra

350 exercises70 sections
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Sections
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Systems of linear equations

47
Geometric view of systems of equations2
Algebraic view of systems of equations2
Elementary operations7
Gaussian elimination27
Gauss-Jordan elimination6
Homogeneous systems2
Fields1

Vectors in Rn

45
Points and vectors2
Addition2
Scalar multiplication2
Linear combinations2
Length of a vector8
The dot product13
The cross product16

Lines and planes in Rn

6
Lines2
Planes4

Matrices

58
Definition and equality4
Addition3
Scalar multiplication2
Matrix multiplication15
Matrix inverses18
Elementary matrices1
The transpose10
Application: Cryptography5

Spans, linear independence, and bases in Rn

28
Spans5
Linear independence9
Subspaces of R^n5
Basis and dimension3
Column space, row space, and null space of a matrix6

Linear transformations in Rn

22
Linear transformations4
The matrix of a linear transformation6
Geometric interpretation of linear transformations2
Properties of linear transformations10

Determinants

35
Minors and cofactors5
The determinant of a triangular matrix1
Determinants and row operations1
Properties of determinants13
Application: A formula for the inverse of a matrix10
Application: Cramer’s rule5

Eigenvalues, eigenvectors, and diagonalization

23
Eigenvectors and eigenvalues6
Finding eigenvalues1
Geometric interpretation of eigenvectors4
Diagonalization3
Application: Matrix powers3
Application: The matrix exponential1
Properties of eigenvectors and eigenvalues2
The Cayley-Hamilton Theorem1
Complex eigenvalues and eigenvectors2

Vector spaces

14
Definition of vector spaces1
Linear combinations, span, and linear independence4
Subspaces3
Basis and dimension5
Application: Error correcting codes1

Linear transformation of vector spaces

12
Definition and examples3
The algebra of linear transformations1
Linear transformations defined on a basis2
The matrix of a linear transformation6

Inner product spaces

53
Real inner product spaces6
Orthogonality6
The Gram-Schmidt orthogonalization procedure7
Orthogonal projections and Fourier series4
Application: Least squares approximations and curve fitting4
Orthogonal functions and orthogonal matrices3
Diagonalization of real symmetric matrices2
Positive semidefinite and positive definite matrices4
Application: Simplification of quadratic forms5
Complex inner product spaces5
Unitary and hermitian matrices7

Complex numbers

7
The complex numbers3
The fundamental theorem of algebra4

Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.