--- title: Linear Independence description: Linear independence describes a set of vectors where no vector can be written as a linear combination of the others. You will learn to determine independence using determinants and row reduction, which is essential for identifying bases in vector spaces. category: mathematics subcategory: linear-algebra difficulty: beginner, intermediate, advanced url: /subject/linear-independence --- # Linear Independence Linear independence describes a set of vectors where no vector can be written as a linear combination of the others. You will learn to determine independence using determinants and row reduction, which is essential for identifying bases in vector spaces. ## Available Resources 1 Videos • 4 Courses • 3 Websites ## Websites ### 1. Khan Academy - Linear Independence Khan Academy lesson from its linear algebra course, using Sal Khan's videos to define span and linear independence and to test whether vector sets are dependent. Learners can decide independence by solving homogeneous systems and relate it to basis and dimension. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/linear-independence **Tags:** linear-algebra, linear-independence, span, vector-spaces ### 2. A First Course in Linear Algebra — Section LI: Linear Independence **Author:** Robert A. Beezer The linear independence section of Beezer's open textbook: complete definitions, proved theorems linking independence to homogeneous systems and reduced row-echelon form, worked examples, Sage verification, reading questions and exercises with solutions. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://linear.pugetsound.edu/html/section-LI.html **Tags:** linear-algebra, linear-independence, homogeneous-systems, row-reduction, proofs ### 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. 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. Independence, Basis and Dimension (MIT 18.06SC) **Author:** Gilbert Strang Defines linear independence through the nullspace of A, then develops spanning sets, basis and dimension as one argument. The session includes the lecture video, summary notes, a recitation video and practice problems with solutions, so learners can test independence and find bases. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/pages/ax-b-and-the-four-subspaces/independence-basis-and-dimension/ **Tags:** linear-independence, basis, dimension, vector-spaces, nullspace ### 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 --- *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-independence