--- title: Linear Transformations description: Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication. You will understand how to represent these mappings using matrices, find their kernel and image, and analyze their geometric effects on space. category: mathematics subcategory: linear-algebra difficulty: beginner, intermediate, advanced url: /subject/linear-transformations --- # Linear Transformations Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication. You will understand how to represent these mappings using matrices, find their kernel and image, and analyze their geometric effects on space. ## Available Resources 2 Videos • 1 Books • 4 Courses • 2 Websites ## Courses ### 1. Essence of Linear Algebra Grasp determinants with 3Blue1Brown's "Essence of Linear Algebra" course. Visualize and understand this key linear algebra concept. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab ### 2. Linear Algebra (MIT 18.06) **Author:** Gilbert Strang Matrix theory and linear algebra: systems of equations, elimination, vector spaces and subspaces, orthogonality and least squares, determinants, eigenvalues and positive definite matrices. Includes 35 lecture videos plus problem sets and exams with solutions, giving the fluency needed for applied mathematics, engineering and data science. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ **Tags:** linear-algebra, vector-spaces, eigenvalues, least-squares, matrix-factorization ### 3. Linear Transformations and Their Matrices (MIT 18.06SC) **Author:** Gilbert Strang Shows how a linear transformation acquires a matrix once bases are chosen for the input and output spaces, with projections, rotations and differentiation as examples. The session includes the lecture video, summary notes, a recitation video and a problem set with solutions. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/pages/positive-definite-matrices-and-applications/linear-transformations-and-their-matrices/ **Tags:** linear-transformations, linear-algebra, matrix-representation, change-of-basis, projections ### 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. Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra **Author:** Grant Sanderson Chapter 2 of 3Blue1Brown's Essence of Linear Algebra series, a short animated lesson on linear combinations, span and basis vectors in 2D and 3D. Viewers learn to picture which points a set of vectors can reach and what linear dependence means geometrically. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab **Tags:** linear-algebra, vectors, linear-combinations, span, basis-vectors ### 2. 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. Hefferon's Linear Algebra (Fourth Edition) **Author:** Jim Hefferon Free proof-oriented undergraduate text whose entire Chapter Three, 'Maps Between Spaces', is devoted to linear maps: isomorphisms first, then homomorphisms, range space and null space, matrix representation, and change of basis. Full worked answers to every exercise. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://hefferon.net/linearalgebra/ **Tags:** linear-algebra, linear-maps, vector-spaces, change-of-basis, proofs, open-textbook ## Websites ### 1. Interactive Linear Algebra — Chapter 3: Linear Transformations and Matrix Algebra **Author:** Dan Margalit and Joseph Rabinoff Georgia Tech's Math 1553 text, with draggable WebGL demonstrations. Chapter 3 works through matrix transformations, one-to-one and onto maps, the proof that every linear transformation is a matrix transformation, composition, inverses, and the Invertible Matrix Theorem. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://textbooks.math.gatech.edu/ila/chap-matrices.html **Tags:** linear-algebra, linear-transformations, matrix-algebra, invertible-matrix-theorem, interactive-textbook ### 2. 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 --- *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/linear-transformations