Northeastern University · Department of Mathematics

MATH 5110

Applied Linear Algebra & Matrix Analysis

A graduate-level treatment of theory, computation, and applications

Instructor: He Wang
Northeastern University, Dept. of Mathematics
hewang.sites.northeastern.edu
About the Course

Course Overview

Rigorous foundations. Comprehensive treatment of linear algebra concepts and computational techniques, well beyond undergraduate level.
Broad applications. Covers key applications in CS, statistics, engineering, machine learning, and data analysis.
Audience. Graduate students in Applied Mathematics, CS, Engineering, and anyone seeking deeper mathematical insight.
Foundation for more. Solid grounding for advanced study in optimization, spectral methods, signal processing, and ML theory.
Visual Roadmap

Complete Linear Algebra Knowledge Map

Root / keystone
Foundations & structure
Operations & algorithms
Eigentheory & decompositions
Discrete applications
Geometry & inner products
Data & ML applications
Advanced / extra topics

✦ Click any node to open a Claude explanation in a new tab

All topics — flow top to bottom, left branch = discrete applications, right branch = geometry & data
Linear system Matrix algebra sum · product · inverse · transpose Vector spaces sum · scalar product · linear maps More matrix operations row ops · LU · trace · det · rank Basis · coordinate · dimension rank-nullity · change of basis Eigenvalues and eigenspaces Diagonalization · Jordan canonical form algebraic / geometric multiplicity · Cayley–Hamilton Markov chains · Perron–Frobenius stochastic matrix · dynamical systems Inner product space norm · Cauchy–Schwarz · geometry Google PageRank algorithm Orthogonal projection orthogonal basis Gram–Schmidt QR factorization Discrete dynamical systems General least squares FFT · Haar wavelets Spectral decomposition SVD · PCA singular values · data compression · pseudo-inverse Image compression ML · data analysis · regression ── ADVANCED & EXTRA TOPICS ── Norms · matrix exponential Spectral graph theory QR algorithm Linear programming Quaternions · rotations Exterior · tensor algebra Matrix calculus · backprop Affine spaces · dual norms Hadamard matrices Orthogonal · unitary groups Chain complex · homology Top 10 algorithms
Curriculum

Main Course Topics

01
Linear Systems & Matrix Algebra: solutions, row reduction, RREF, inverses, transpose
Ask Claude to explain
02
Linear Spaces: subspaces, basis & dimension, rank-nullity theorem, linear transformations, change of basis
Ask Claude to explain
03
Jordan Normal Forms: determinants, eigenvalues & eigenvectors, diagonalization, Cayley–Hamilton
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04
Inner Product Spaces: Cauchy–Schwarz, Gram–Schmidt, positive definite, SVD
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05
PCA: covariance matrices, optimal linear combinations, data compression
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06
Markov Chains & Perron–Frobenius: stochastic matrices, dynamical systems
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07
Fourier Transform & FFT, Haar Wavelets, Hadamard matrices
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08
Norms & Matrix Norms: convergence, matrix exponential eAt
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09
Algebraic & Spectral Graph Theory: Laplacian, connectivity, graph spectra
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10
QR Algorithm: numerical eigenvalue computation, iterative methods
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11
Quaternions & Rotations: SU(2), SO(3), unitary groups U(n)
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12
Linear Programming, Matrix Calculus, Neural Networks & Backpropagation
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13
Exterior Algebra, Tensor Algebra, Multilinear Algebra
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14
Affine Spaces & Affine Maps, Dual Norms, matrix inequalities
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Quick Reference

Key Concepts — Ask Claude

Matrix Factorizations
LU decomposition QR decomposition SVD Jordan form Schur decomposition Cholesky
Eigentheory
Eigenvalues & Eigenvectors Diagonalization Spectral Theorem Cayley–Hamilton Characteristic Polynomial Minimal Polynomial
Geometry & Inner Products
Inner Product Space Gram–Schmidt Orthogonal Projection Cauchy–Schwarz Positive Definite Hilbert Space
Applications
PCA Markov Chains Google PageRank Least Squares FFT SVM Perron–Frobenius Backpropagation
Advanced Topics
Tensors Exterior Algebra Matrix Exponential Quaternions Spectral Graph Theory Linear Programming Dual Norms Wavelets
Bibliography

References & Textbooks

Perspective

A Note on ML Jargon

Jean Gallier Q&A

Q: Do support vector machines chase cattle to catch them with some kind of super lasso?

A: No. Behind the jargon lies a lot of "classical" linear algebra and techniques from optimization theory — and probability theory and statistics.

Ask Claude: kernel PCA ridge / lasso SVM Lagrange / KKT