--- title: Applications of Linear Algebra description: This topic explores how linear algebra solves real-world problems across science and engineering. Learners will understand how matrices and vector spaces are applied in computer graphics, cryptography, search engines, and machine learning. category: mathematics subcategory: linear-algebra difficulty: beginner, intermediate, advanced url: /subject/applications-of-linear-algebra --- # Applications of Linear Algebra This topic explores how linear algebra solves real-world problems across science and engineering. Learners will understand how matrices and vector spaces are applied in computer graphics, cryptography, search engines, and machine learning. ## Available Resources 1 Videos • 1 Books • 4 Courses • 3 Websites ## Websites ### 1. 3Blue1Brown Linear Algebra Series Grant Sanderson's animated series on the geometric meaning of linear algebra: vectors, linear transformations, matrix multiplication, determinants, eigenvectors and abstract vector spaces. Viewers come away able to picture what a matrix does to space, which makes later computational courses far easier to follow. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.3blue1brown.com/topics/linear-algebra **Tags:** linear-algebra, linear-transformations, eigenvectors, determinants, geometric-intuition ### 2. Mathematics for Machine Learning A free textbook by Deisenroth, Faisal and Ong (Cambridge University Press) covering the linear algebra, analytic geometry, matrix decompositions, vector calculus and probability behind machine learning, then derives linear regression, PCA, Gaussian mixtures and support vector machines from them. **Difficulty:** Beginner | **Price:** Free **Link:** https://mml-book.github.io/ **Tags:** linear-algebra, machine-learning, matrix-decompositions, pca, vector-calculus ### 3. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Courses ### 1. Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (MIT 18.065) **Author:** Gilbert Strang Linear algebra for data science and deep learning: singular value decomposition, low-rank approximation, least squares, PCA, randomized linear algebra, gradient descent and the structure of neural networks. Includes 37 lecture videos and problem sets. Afterwards you can recognise the matrix computations inside modern machine learning. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/ **Tags:** singular-value-decomposition, matrix-factorization, principal-component-analysis, optimization, deep-learning ### 2. Computational Science and Engineering I (MIT 18.085) **Author:** Gilbert Strang Applied linear algebra for networks, structures and estimation, followed by equilibrium differential equations, Laplace's equation, boundary-value problems, calculus of variations, Fourier series and the discrete Fourier transform. Includes 50 lecture videos, problem sets and exams with solutions, and programming assignments. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-fall-2008/ **Tags:** applied-linear-algebra, boundary-value-problems, calculus-of-variations, fourier-analysis, differential-equations ### 3. Linear Algebra - Foundations to Frontiers Learn the mathematics behind linear algebra and link it to matrix software development. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.edx.org/learn/linear-algebra/the-university-of-texas-at-austin-linear-algebra-foundations-to-frontiers **Tags:** courses, mathematics-statistics, linear-algebra ### 4. Mathematics for Machine Learning: Linear Algebra In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets - like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works. Since we're aiming at data-driven applications, we'll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you'll write code blocks and encounter Jupyter notebooks in Python, but don't worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before. At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these concepts to machine learning. **Difficulty:** Beginner | **Language:** English | **Duration:** 5 weeks of study, 2-5 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/linear-algebra-machine-learning **Tags:** courses, mathematics-statistics, linear-algebra ## Videos ### 1. Vectors | Chapter 1, Essence of Linear Algebra **Author:** Grant Sanderson Opening chapter of Grant Sanderson's animated linear algebra series, contrasting the physics, computer science and mathematics views of vectors. Shows vectors as arrows and coordinate lists, and how addition and scalar multiplication work geometrically, laying intuition for linear combinations and transformations. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/watch?v=fNk_zzaMoSs **Tags:** linear-algebra, vectors, vector-addition, visual-intuition ## Books ### 1. Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares (free PDF) **Author:** Stephen Boyd, Lieven Vandenberghe Stanford and UCLA authors' applied linear algebra text, free PDF from Cambridge. Covers solving square and overdetermined systems via QR factorization and back substitution, with operation counts, so you understand what a solver actually does and when least squares replaces exact solution. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://web.stanford.edu/~boyd/vmls/ **Tags:** linear-algebra, least-squares, qr-factorization, matrices, applied-math --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/applications-of-linear-algebra