Applied Linear Algebra

Applied Linear Algebra Class Notes

Keywords: Applied Linear Algebra notes, Olver and Shakiban, eigenvalues, Gram–Schmidt, orthogonal projections, singular values, vector spaces.

As an instructor of applied linear algebra, I prepared these class notes as a review of the chapters in

“Applied Linear Algebra, 2nd Edition” by Olver and Shakiban. They include example problems which have been worked out in some detail. I hope you find these useful.

Class Notes
1.1–1.3 Linear Systems, Matrices, Vectors, Gaussian Elimination
1.4–1.5 Pivoting, Permutations, Inverse Matrices
1.6 Transposes and Symmetric Matrices
1.8 General Linear Systems
1.9 Determinants
2.1 Real Vector Spaces
2.2 Subspaces
2.3 Span and Linear Independence
2.4 Basis and Dimension
2.5 Fundamental Matrix Subspaces
2.6 Graphs and Digraphs
3.1 Inner Products
3.2 Cauchy–Schwarz and Triangle Inequalities
3.3 Norms
3.4 Positive Definite Matrices
3.5 Completing the Square
4.1 Orthogonal and Orthonormal Bases
4.2 Gram–Schmidt Process
4.3 Orthogonal Matrices
4.4 Orthogonal Projections and Subspaces; Fredholm Alternative
5.2 Minimization of Quadratic Functions
7.1 Linear Functions
7.2 Linear Transformations
8.2 Eigenvalues and Eigenvectors
8.3 Eigenvector Bases
8.5 Eigenvalues of Symmetric Matrices
8.7 Singular Values