Unit 1
Vectors, matrices, and linear maps
Vector data, dot products, cosine similarity, matrix-vector products, matrix multiplication, and affine maps.
Fall 2026
MATH 345: Linear algebra and optimization
A first course in linear algebra with selected multivariable calculus, optimization, AI & data science applications, and Python coding.
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Course information
MATH 345 introduces vectors, matrices, systems of linear equations, linear maps, subspaces, bases, rank, orthogonality, projection, least squares, partial derivatives, gradients, Jacobians, local linearization, gradient descent, eigenvalues, Hessians, Lagrange multipliers, principal component analysis, singular value decomposition, low-rank approximation, polynomial approximation, and Python implementation.
The course is intended for students who want a first linear algebra course that also develops optimization and computational tools used in data science and related fields.
Archive: Spring 2025 website
Prerequisites and credits
Who should consider MATH 345? MATH 345 is a first linear algebra course with optimization and Python-based computation. The course is especially recommended for students who do not plan to take MATH 234 but wish to learn about multivariate optimization as preparation for AI and machine learning courses. Students who have taken or plan to take MATH 234 should check advising guidance before choosing among MATH 340, MATH 341, and MATH 345. Students who want a more proof-focused linear algebra course may want to consider MATH 341.
Note on the number of credits: While MATH 320 and MATH 340 are only 3 credits (for historical reasons), the number of class/discussion hours and the amount of work required are meant to be roughly the same as MATH 345.
Topics
Unit 2
Anatomy of a linear map
Linear systems, row reduction, solution sets, subspaces, null spaces, image spaces, bases, rank, inverses, and determinants.
Unit 3
Local linearization and first-order optimization
Functions of several variables, limits, continuity, partial derivatives, gradients, Hessians, Jacobians, local linearization, the chain rule, directional derivatives, and gradient descent.
Unit 4
Orthogonality and least squares
Orthogonal bases, projections, Gram–Schmidt, least squares, QR factorization, regression, and residuals.
Unit 5
Eigenvalues and second-order optimization
Eigenvalues, eigenvectors, diagonalization, symmetric matrices, quadratic forms, definiteness, Hessians, the second derivative test, and least squares with fixed nonlinear features.
Unit 6
Constrained optimization and Lagrange multipliers
Absolute extrema, Lagrange multipliers, quadratic forms, maximum stretch, principal component analysis (PCA), singular value decomposition (SVD), and low-rank approximation.
Unit 7
Abstract vector spaces
Abstract vector spaces, inner product spaces, polynomial spaces, continuous least squares, Taylor approximation, and Legendre polynomials.
Online Lecture Notes
The textbook contains the main exposition, activities, definitions, theorems, examples, review checkpoints, and solutions. It is designed for web reading, with expandable solutions and cross-references.
- Activity-focused treatment of linear algebra, differential calculus of several variables, optimization, least squares, and SVD.
- Python labs linked to the mathematical material.
- Accessible HTML output with expandable solutions and cross-references.
Read the textbook
Archive: Spring 2026 notes