Applied Linear Algebra Notes and Python Practice Activities
As an instructor of applied linear algebra, I prepared these materials to accompany
topics from
Applied Linear Algebra, 2nd Edition by Olver and Shakiban
.
The page includes class notes and worked examples, together with Python notebooks,
Doenet practice files, and group worksheets modeled on the kinds of activities used
in class.
The notes are organized by textbook topic. The computational activities are organized
as a separate ten-part sequence, while the group worksheets are kept in their own
section because they often combine ideas from several nearby lessons.
Linear Systems, Matrices, and Determinants
Foundational material on linear systems, matrices, vectors, Gaussian elimination, inverses, transposes, symmetric matrices, and determinants.
Vector Spaces and Subspaces
Core vector-space topics, including subspaces, span, linear independence, bases, dimension, and fundamental matrix subspaces.
Inner Products, Norms, and Quadratic Forms
Inner products, Cauchy-Schwarz, triangle inequalities, norms, positive definite matrices, and completing the square.
Orthogonality and Projections
Orthogonal and orthonormal bases, the Gram-Schmidt process, orthogonal matrices, projections, subspaces, and the Fredholm alternative.
Linear Functions and Linear Transformations
Linear functions and linear transformations, emphasizing structure, mappings, and matrix representations.
Eigenvalues, Eigenvectors, and Singular Values
Eigenvalues, eigenvectors, eigenvector bases, symmetric matrices, and singular values.
Python Notebooks and Interactive Doenet Practice
These activities are organized as a ten-part practice sequence. The Python notebooks
provide computational exploration and numerical practice. The corresponding Doenet
materials provide guided, interactive mathematical practice on the same broad topics.
They are intended to reinforce the mathematical ideas in the notes rather than replace
the written work.
Doenet is an open-source system for creating interactive mathematics
and other educational activities. Its DoenetML language can display mathematics, accept
student input, check answers, and build interactive components. Because Doenet is less
familiar than systems such as Jupyter or Google Colab, instructions for using the files
are included below.
The .ipynb files are Jupyter notebooks. The .doenet files are
DoenetML source files; depending on your browser, clicking one may download the source
rather than display an activity directly.
How to Use the Computational and Interactive Materials
Practice 1: Vectors and Matrices
Use Python and guided practice to work with vectors, matrices, dimensions, and basic matrix operations.
Practice 2: Systems and Residuals
Connect linear systems with computational solutions and use residuals to measure how well a proposed solution satisfies a system.
Practice 3: Determinants and Transposes
Practice transpose operations and determinant calculations while connecting them to matrix structure and invertibility.
Practice 4: Fundamental Subspaces
Explore column spaces, row spaces, null spaces, rank, and the relationships among the fundamental subspaces of a matrix.
Practice 5: Inner Products and Quadratic Forms
Work with inner products, norms, orthogonality, and quadratic forms in both symbolic and computational settings.
Practice 6: Gram-Schmidt and QR
Carry out Gram-Schmidt orthogonalization and connect the process to QR factorization.
Practice 7: Least Squares
Use orthogonal projection and matrix methods to study inconsistent systems and least-squares approximations.
Practice 8: Change of Basis
Practice coordinate vectors, basis matrices, and moving representations of vectors and linear maps from one basis to another.
Practice 9: Complex Eigenvalues and Dynamics
Explore complex eigenvalues and eigenvectors and how eigenstructure helps describe the behavior of linear dynamical systems.
Practice 10: Symmetric Eigenvalues and SVD
Study the special eigenvalue structure of symmetric matrices and connect it with singular values and the singular value decomposition.
Group Worksheets and Quiz-Style Review
These worksheets were developed for collaborative recitation work and quiz preparation.
They are deliberately listed separately from the ten practice lessons: a worksheet may
combine material from several nearby lessons, so its number identifies the worksheet
sequence and should not be read as a one-to-one match with a particular lesson above.
A useful approach is to work through the notes and practice activities first, then use
the worksheets for mixed review once several related ideas have been introduced.
