LEARN / LINEAR ALGEBRA
Linear algebra, learned well.
A short, opinionated list of the best linear algebra material in English and Chinese, with a path from first intuition to proofs and real applications. Nearly everything here is free.
A suggested path
Most people get stuck because they learn the mechanics before the meaning. Do it the other way around.
- 01Build intuition first
Watch 3Blue1Brown’s Essence of Linear Algebra. In about a weekend you will see vectors, matrices, determinants, and eigenvectors as geometry rather than symbols.
- 02Take one complete course
Follow MIT 18.06 with Gilbert Strang from start to finish. If you study for a Chinese university exam, Song Hao’s course matches that syllabus closely.
- 03Do the exercises
Work through the problem sets after each lecture. Watching without solving problems does not stick. Hefferon’s free book includes worked answers to every exercise.
- 04Learn the theory
Read Linear Algebra Done Right when you want proofs and abstract vector spaces. It is the standard second course and is free in English and Chinese.
- 05Put it to work
Use Boyd and Vandenberghe’s applied book, MIT 18.065, or Mathematics for Machine Learning to apply least squares, SVD, and PCA to real data.
Build intuition
Start here, even if you have taken a course before.
Essence of Linear Algebra
Grant Sanderson (3Blue1Brown)
Short animated videos that show what vectors, matrix multiplication, determinants, change of basis, and eigenvectors mean geometrically. The best first step there is.
Immersive Linear Algebra
J. Ström, K. Åström, T. Akenine-Möller
A complete textbook whose figures you can drag and rotate in the browser. Good for seeing 3D ideas such as cross products and projections.
线性代数的本质 (Chinese subtitles)
3Blue1Brown’s official Bilibili channel
The same series with Chinese and English subtitles, on 3Blue1Brown’s official Bilibili channel. Easy to reach from mainland China.
Complete courses
Pick one and finish it. Each covers a full first course.
MIT 18.06SC Linear Algebra
Gilbert Strang · MIT OpenCourseWare
The most famous linear algebra course in the world: Strang’s lectures, plus recitation videos, problem sets, and exams with solutions. Pairs with his textbook.
Khan Academy Linear Algebra
Sal Khan · Khan Academy
Patient, step-by-step videos on vectors and spaces, matrix transformations, and alternate bases. Good when you need a slower explanation of one topic.
MIT 18.06 with Chinese subtitles
Gilbert Strang · NetEase Open Courses
Strang’s complete 18.06 lectures with Chinese and English subtitles, hosted by NetEase Open Courses.
Linear Algebra with Song Hao
Song Hao · Bilibili
Clear, exam-focused lectures in Chinese that follow the mainland university syllabus: determinants, matrices, vectors, linear systems, eigenvalues, and quadratic forms.
Linear Algebra 2018 (NTU)
Hung-yi Lee · National Taiwan University
A full university course taught in Mandarin with English slides, with a strong eye on applications and Python homework. Videos are on YouTube.
Textbooks
One main book is enough. The free ones are complete and legal to download.
Linear Algebra Done Right, 4th edition
Sheldon Axler
The standard proof-based second course, built around linear maps, with determinants saved for the end. The author offers free PDFs in English, Chinese, and other languages.
Introduction to Linear Algebra, 6th edition
Gilbert Strang
The book behind MIT 18.06. The book is paid, but its website has sample sections, solutions, and links to every lecture.
Interactive Linear Algebra
Dan Margalit, Joseph Rabinoff · Georgia Tech
A free online textbook with interactive demonstrations throughout. Strong on geometric pictures of solution sets and transformations.
Linear Algebra
Jim Hefferon
A careful, free textbook that bridges computation and proof, with complete worked answers to every exercise. Ideal for self-study.
Engineering Mathematics: Linear Algebra
Tongji University · Higher Education Press
The most widely used linear algebra textbook in Chinese engineering programs, and the usual reference for the graduate entrance exam.
Advanced Algebra
Peking University · Higher Education Press
The classic text for mathematics majors in China, covering polynomials, linear algebra, and lambda-matrices with more rigor.
Applications: data science and machine learning
For when you want to use linear algebra, not only pass an exam.
Introduction to Applied Linear Algebra (VMLS)
Stephen Boyd, Lieven Vandenberghe
Vectors, matrices, and least squares, taught through real applications such as regression, classification, and control. Free PDF with Python and Julia companions.
MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
Gilbert Strang · MIT OpenCourseWare
Strang’s follow-up to 18.06: SVD, low-rank approximation, optimization, and the linear algebra inside deep learning.
Mathematics for Machine Learning
Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong
Its chapters on linear algebra, analytic geometry, and matrix decompositions are a compact refresher aimed squarely at machine learning.
Dive into Deep Learning: Linear Algebra
Aston Zhang, Zachary Lipton, Mu Li, Alexander Smola
A hands-on chapter that shows scalars, vectors, matrices, tensors, and norms as runnable code. Available in English and Chinese.
Computational Linear Algebra
Rachel Thomas · fast.ai
How linear algebra is actually computed: fast and accurate matrix methods, randomized SVD, and NMF, taught in Jupyter notebooks.
What you should be able to do
Use this as a checklist. Every resource below covers these ideas, in different orders and with different emphasis.
- Vectors, linear combinations, span, linear independence, basis, and dimension
- Matrices as linear transformations, and matrix multiplication as composition
- Gaussian elimination, LU factorization, rank, and the four fundamental subspaces
- Solving Ax = b, inverse matrices, and when a solution exists or is unique
- Determinants as signed volume, and their properties
- Orthogonality, projections, least squares, Gram–Schmidt, and QR
- Eigenvalues, eigenvectors, and diagonalization
- Symmetric and positive definite matrices, and quadratic forms
- The singular value decomposition (SVD) and principal component analysis (PCA)
Common questions
I have never studied linear algebra. Where should I start?
Watch 3Blue1Brown’s Essence of Linear Algebra first, then take MIT 18.06 with Gilbert Strang and do its problem sets. That combination works for most beginners.
Which resources fit a Chinese university exam or the graduate entrance exam?
Song Hao’s video course with the Tongji University textbook matches the mainland syllabus. Add 3Blue1Brown for intuition, so that the computations make sense.
Which resources are best for machine learning?
After a first course, read Introduction to Applied Linear Algebra (VMLS) or the linear algebra chapters of Mathematics for Machine Learning, then watch MIT 18.065 for SVD and low-rank methods.
Are these resources free?
Every linked resource is free to use, except Strang’s textbook, whose companion site is free. The Tongji and Peking University books are print books. Some videos are on YouTube, which may be blocked in mainland China, so Chinese alternatives on Bilibili and NetEase are listed too.