Linear combinations, Ax as a mix of columns, the column space, row and column pictures, matrices that move the plane, and why AB is not BA. 8 short slides, then worked examples and problems. Each slide has an everyday picture, a worked example and Python you can run right on the page.
What cv + dw really means, with an everyday picture, a worked example and Python you can run in your browser. Follows Strang §1.1.
02 · Strang §1.3 · MatricesWhy Ax is a mix of the columns of A, explained with a shopping basket, a worked example and live Python. Row way vs column way. Strang §1.3.
03 · Strang §1.3 and §3.1 · Column spaceThe column space is everything Ax can reach. Learn it with a two-thruster boat, rank, a worked example and Python you can run. Strang §1.3 and §3.1.
04 · Strang §2.1 · Vectors and linear equationsTwo ways to see a linear system: lines crossing, or columns combining to make b. Interactive plot and Python in the browser. Strang §2.1.
05 · Strang §8.1 · Linear transformationsSee a matrix move the plane. Why the columns are where (1, 0) and (0, 1) land, how a rotation matrix is built, and what the determinant measures.
06 · Strang §2.4 · Rules for matrix operationsAB means apply B first, then A. See it with a rotation and a shear, learn the rules of matrix products, and try it in Python. Strang §2.4.
07 · ChallengeBuild the 2x2 matrix that mirrors the plane, and see why a reflection has determinant −1. An interactive linear algebra challenge in Python.
08 · Recap · ReadingSix core ideas of linear algebra on one page: combinations, Ax by columns, column space, row and column pictures, transformations and matrix products.
✎ · PracticeThree worked examples and five practice problems with hidden solutions on linear combinations, Ax by columns, column space, transformations and matrix products.