--- title: Vector Spaces description: Vector spaces are algebraic structures consisting of vectors that can be added and multiplied by scalars. Learners will understand linear independence, bases, dimension, and linear transformations, enabling them to solve systems of linear equations. category: mathematics subcategory: abstract-algebra difficulty: beginner, intermediate, advanced url: /subject/vector-spaces --- # Vector Spaces Vector spaces are algebraic structures consisting of vectors that can be added and multiplied by scalars. Learners will understand linear independence, bases, dimension, and linear transformations, enabling them to solve systems of linear equations. ## Available Resources 3 Videos • 1 Books • 5 Courses • 2 Websites ## Videos ### 1. Linear transformations and matrices | Essence of linear algebra, chapter 3 3Blue1Brown video explaining how matrices encode linear transformations of vector spaces, showing how basis vectors determine a transformation and how matrix multiplication corresponds to composing transformations. Part of the Essence of Linear Algebra series. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/watch?v=kYB8IZa5AuE **Tags:** linear-algebra, video, visualization ### 2. 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 ### 3. 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 ## Courses ### 1. Linear Algebra - Khan Academy Learn linear algebra fundamentals, including matrices, vectors, and transformations, with Khan Academy's comprehensive course. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/linear-algebra **Tags:** linear-algebra, matrices, vectors, linear-transformations, eigenvalues ### 2. 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 **Tags:** determinants, linear-algebra, linear-transformations, geometric-intuition ### 3. 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 ### 4. Linear Algebra This course is an introduction to linear algebra. You will discover the basic objects of linear algebra – how to compute with them, how they fit together theoretically, and how they can be used to solve real problems. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.edx.org/learn/linear-algebra/rice-university-linear-algebra-part-1 **Tags:** courses, mathematics-statistics, abstract-algebra ### 5. 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 ## Websites ### 1. Vector Spaces on nLab nLab reference article defining vector spaces as modules over a field and framing them within category theory via the category Vect_k of vector spaces and linear maps. Useful for readers wanting the categorical view of linear algebra. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/vector+space **Tags:** linear-algebra, category-theory, reference ### 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 ## Books ### 1. Linear Algebra Done Right **Author:** Sheldon Axler Axler's widely used linear algebra textbook develops the subject through vector spaces and linear operators, deferring determinants until near the end. Covers eigenvalues, inner product spaces, and the spectral theorem, building conceptual understanding valued in later coursework in analysis, physics, and algebra. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387982582?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, abstract-algebra --- *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/vector-spaces