--- title: Linear Algebra description: Linear algebra is the study of vectors, vector spaces, and linear transformations represented by matrices. Learners will understand how to solve systems of linear equations, compute eigenvalues, and apply these concepts to data science. category: mathematics subcategory: linear-algebra difficulty: beginner, intermediate, advanced url: /subject/linear-algebra --- # Linear Algebra Linear algebra is the study of vectors, vector spaces, and linear transformations represented by matrices. Learners will understand how to solve systems of linear equations, compute eigenvalues, and apply these concepts to data science. ## Available Resources 3 Videos • 4 Books • 18 Courses • 3 Websites ## Books ### 1. Applied Linear Algebra (Boyd & Vandenberghe) **Author:** Stephen Boyd, Lieven Vandenberghe Modern linear algebra presented as the essential infrastructure for data science and AI optimization. **Difficulty:** Intermediate | **Price:** Paid **Link:** https://www.amazon.com/s?k=1108416410&tag=edmonddante07-20 **Tags:** linear-algebra, least-squares, matrices, applied-mathematics, optimization ### 2. Linear Algebra Done Right **Author:** Sheldon Axler Rigorous linear algebra text emphasizing vector spaces, linear maps, and abstract structure over computational techniques and determinants. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://linear.axler.net/ **Tags:** book, mathematics, algebra ### 3. Linear Algebra and Its Applications **Author:** David C. Lay Undergraduate linear algebra textbook that builds from row reduction of linear systems through matrix algebra, determinants, vector spaces, eigenvalues, orthogonality, and least squares, with applications from economics, engineering, and computer graphics. Readers learn to solve and analyze systems of equations using matrices. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0201824787?tag=edmonddante07-20 **Tags:** linear-algebra, matrices, linear-systems, eigenvalues, vector-spaces, least-squares ### 4. No Bullshit Guide to Linear Algebra **Author:** Ivan Savov Compact textbook that rebuilds vectors, matrices, determinants, eigendecomposition and vector spaces starting from high-school algebra, then applies them to cryptography, error-correcting codes, quantum mechanics and graphics. Readers finish able to compute by hand and follow the proofs. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0992001021?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, mathematics ## Courses ### 1. A Vision of Linear Algebra (MIT RES.18-010) **Author:** Gilbert Strang Ten short lecture videos with slides presenting a recommended order for linear algebra: column space, the four fundamental subspaces, orthogonality, eigenvalues, singular values, least squares, and five matrix factorizations including A = CR. Gives a big-picture review connecting the core ideas. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-18-010-a-2020-vision-of-linear-algebra-spring-2020/ **Tags:** linear-algebra, matrix-factorization, four-fundamental-subspaces, eigenvalues, singular-value-decomposition ### 2. 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 ### 3. Linear Algebra (MIT 18.06SC) **Author:** Gilbert Strang Matrix theory and linear algebra following Strang's textbook: elimination, vector spaces, orthogonality and least squares, determinants, eigenvalues, positive definite matrices and the singular value decomposition. Includes 74 lecture and problem-solving videos, summary notes, and problem sets and exams with solutions, built for independent study. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/ **Tags:** linear-algebra, vector-spaces, eigenvalues, least-squares, singular-value-decomposition ### 4. 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 ### 5. Introduction to Functional Analysis (MIT 18.102) **Author:** Casey Rodriguez Analysis on infinite-dimensional spaces: normed and Banach spaces, the Hahn-Banach theorem and duality, bounded operators, Lebesgue measure and Lp spaces, Hilbert spaces, compact and self-adjoint operators, and the spectral theorem. Includes 23 lecture videos, extensive lecture notes, problem sets and exams. **Difficulty:** Advanced | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/ **Tags:** functional-analysis, banach-spaces, hilbert-spaces, spectral-theorem, operator-theory, lebesgue-integration ### 6. Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler (MIT RES.18-009) **Author:** Gilbert Strang, Cleve Moler Short video lessons on ordinary differential equations, from first-order and second-order equations to Laplace transforms, linear systems, eigenvalues and matrix exponentials, alongside demonstrations of numerical solution with the MATLAB ODE suite. Includes 68 videos pairing analytical methods with practical computation. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-18-009-learn-differential-equations-up-close-with-gilbert-strang-and-cleve-moler-fall-2015/ **Tags:** ordinary-differential-equations, numerical-methods, matlab, laplace-transform, linear-systems ### 7. Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra (MIT RES.18-008) **Author:** Herbert Gross Third part of a 1972 lecture series, covering complex variables, ordinary differential equations and linear algebra at the sophomore level. Provides 20 lecture videos (about 11.5 hours), study guides, supplementary notes and problem sets with solutions. Assumes single and multivariable calculus. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/ **Tags:** differential-equations, complex-variables, linear-algebra, ordinary-differential-equations ### 8. Principles of Applied Mathematics (MIT 18.311) **Author:** Rodolfo Rosales Continuum applied mathematics through examples such as traffic flow, fluids and granular flows: conservation laws, kinematic waves, characteristics and shocks, diffusion, finite differences and stability, and Fourier and spectral methods. Provides 41 lecture note files and problem sets for modeling wave and diffusion problems. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/ **Tags:** partial-differential-equations, conservation-laws, shock-waves, diffusion, numerical-methods ### 9. Differential Equations (MIT 18.03SC) **Author:** Arthur Mattuck, Haynes Miller, Jeremy Orloff, et al. Modeling with ordinary differential equations and solving them: first-order equations, linear second-order equations and oscillations, Fourier series, Laplace transforms, and linear and nonlinear systems in the phase plane. Includes 72 lecture and recitation videos, course notes, and problem sets and exams with solutions. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/ **Tags:** ordinary-differential-equations, laplace-transform, fourier-series, linear-systems, mathematical-modeling ### 10. Multivariable Calculus (MIT 18.02SC) **Author:** Denis Auroux Differential, integral and vector calculus in several variables: vectors and matrices, partial derivatives, optimization, double and triple integrals, line integrals, and the theorems of Green, Stokes and Gauss. Includes 106 lecture and recitation videos, notes, and problems and exams with solutions, built for independent study. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/ **Tags:** multivariable-calculus, partial-derivatives, multiple-integrals, vector-calculus, line-integrals ### 11. Linear Algebra (Full Course) **Author:** Trefor Bazett A one-semester introductory linear algebra video course covering linear systems, row reduction, span, linear independence, linear transformations, bases, dimension, eigenvalues, diagonalization, orthogonality and Gram-Schmidt. Learners can solve systems, reason geometrically about matrices, and follow proofs of core results. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 **Tags:** linear-algebra, matrices, vector-spaces, linear-transformations, eigenvalues ### 12. 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 ### 13. 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 ### 14. Topics in Theoretical Computer Science: An Algorithmist's Toolkit (MIT 18.409) **Author:** Jonathan Kelner Geometric and spectral techniques used in modern algorithm design, starting with spectral graph theory: graph Laplacians, spectral partitioning, Cheeger's inequality, expanders and random walks. 25 lecture-note files and problem sets equip learners to apply eigenvalue methods to algorithmic problems. **Difficulty:** Advanced | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/ **Tags:** spectral-graph-theory, graph-laplacian, cheeger-inequality, expander-graphs, random-walks ### 15. Mathematical Methods for Engineers II (MIT 18.086) **Author:** Gilbert Strang Continuation of 18.085 covering numerical methods for initial-value problems and partial differential equations, finite differences, network flows and optimization. Includes 29 lecture videos, problem sets with solutions and example projects, for learners ready to build and analyze numerical solvers. **Difficulty:** Advanced | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-086-mathematical-methods-for-engineers-ii-spring-2006/ **Tags:** finite-difference-methods, initial-value-problems, partial-differential-equations, numerical-linear-algebra, optimization ### 16. 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 ### 17. Linear Algebra University of Sydney course covering geometric vectors, dot and cross products, systems of equations solved by Gaussian elimination, matrix arithmetic, determinants, eigenvalues and diagonalisation. Finishes with linear transformations and Markov processes, emphasising geometric intuition over formal proof. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/introduction-to-linear-algebra **Tags:** courses, mathematics-statistics, mathematics ### 18. 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 ## Websites ### 1. Khan Academy Linear Algebra: Vectors and Spaces Khan Academy's first linear algebra unit: short videos and exercises on vectors, linear combinations, span, linear independence, subspaces and bases, dot and cross products, solving systems by row reduction, and null and column spaces, building the foundation for matrix transformations. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces **Tags:** linear-algebra, vectors, vector-spaces, span, row-reduction, null-space ### 2. 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 ### 3. Immersive Linear Algebra **Author:** J. Ström, K. Åström, T. Akenine-Möller Interactive linear algebra textbook with dynamic visualizations covering vectors, matrices, transformations, eigenvalues, and decompositions. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://immersivemath.com/ila/index.html **Tags:** website, mathematics, algebra ## Videos ### 1. Introduction to Linear Algebra: Systems of Linear Equations **Author:** Dave Farina A short lecture from Professor Dave Explains introducing linear algebra through systems of linear equations. It shows how such systems are written and solved and how they lead to matrices, giving a first footing before studying matrix methods. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/watch?v=csgNflj69-Y **Tags:** linear-algebra, linear-equations, systems-of-equations, matrices ### 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 ## Youtubes ### 1. 3Blue1Brown **Author:** Grant Sanderson Visual mathematics explanations using animations to build deep intuition for linear algebra, calculus, differential equations, and mathematical concepts. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/@3blue1brown **Tags:** youtube, mathematics, 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/linear-algebra