Linear Algebra

A rigorous graduate-level course on linear algebra—from vector spaces and linear maps through spectral theory and matrix decompositions to advanced applications in data science and functional analysis.

Course Overview

Linear algebra is the backbone of modern mathematics, physics, engineering, and data science. This course develops the subject from abstract vector spaces and linear transformations through eigenvalue theory, canonical forms, and matrix decompositions, culminating in applications to numerical methods and machine learning.

What You'll Learn

  • • Vector spaces, bases, and dimension
  • • Linear maps, matrices, and determinants
  • • Eigenvalues, eigenvectors, and diagonalization
  • • Inner product spaces and the spectral theorem
  • • Jordan normal form and invariant subspaces
  • • Singular value decomposition (SVD)
  • • Tensors and multilinear algebra
  • • Numerical methods and data science applications

Prerequisites

  • • Introductory linear algebra (matrix operations)
  • • Calculus (single and multivariable)
  • • Mathematical proof techniques
  • • Basic set theory and logic

References

  • • S. Axler, Linear Algebra Done Right (4th ed.)
  • • G. Strang, Linear Algebra and Its Applications
  • • R. Horn & C. Johnson, Matrix Analysis
  • • P. Halmos, Finite-Dimensional Vector Spaces

Course Structure

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Part I: Foundations

Vector spaces, linear maps, matrices and determinants, and systems of equations.

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Part II: Spectral Theory

Eigenvalues and eigenvectors, diagonalization, inner product spaces, and the spectral theorem.

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Part III: Decompositions

Jordan normal form, SVD, matrix decompositions, and least squares methods.

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Part IV: Advanced Topics

Tensors and multilinear algebra, functional analysis introduction, numerical methods, and data science applications.

Key Equations

Linear System

$$A\mathbf{x} = \mathbf{b}$$

The fundamental problem of linear algebra: solving systems of linear equations

Determinant

$$\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^{n} a_{i,\sigma(i)}$$

The Leibniz formula for the determinant as a sum over permutations

Eigenvalue Equation

$$A\mathbf{v} = \lambda \mathbf{v}$$

Eigenvectors are stretched by the linear map; eigenvalues give the scaling factor

Singular Value Decomposition

$$A = U \Sigma V^*$$

Every matrix factors into orthogonal rotations and a diagonal scaling

Spectral Theorem

$$A = Q \Lambda Q^{-1}, \quad A = A^* \Rightarrow A = U \Lambda U^*$$

Normal operators are unitarily diagonalizable; self-adjoint operators have real eigenvalues

Jordan Normal Form

$$A = P J P^{-1}, \quad J = \text{diag}(J_1, J_2, \ldots, J_k)$$

The canonical form for any linear operator over an algebraically closed field