This book by Gilbert Strang, author of the widely-used Introduction to Linear Algebra, presents a “new start” in teaching and learning linear algebra, a subject that is increasingly important in all quantitative disciplines. Working with small matrices of integers, students can see dependent columns and the rank of \(A\) and its column space. This is a big step toward the four fundamental subspaces — a widespread success in teaching linear algebra. The new start makes linear algebra more approachable to students with a wide range of backgrounds.

The next step is the column-row factorization \(A = CR\) with independent columns in \(C\). Combinations of those columns produce all columns in \(A\) — this is matrix multiplication by \(R\). The text goes on to solve \(Ax = b\), to orthogonal bases, to subspaces and linear transformations, always based on examples. The final steps are eigenvalues and singular values, with a dedicated code to compress images. The last chapter is entirely optional: linear algebra in deep learning and AI. The text features many illustrative examples and exercises, and video lectures to accompany the book are available via MIT OpenCourseWare. The author's site, contains additional sample material and an instructor's manual.