Subspaces, the nullspace, pivot and free columns, the complete solution to Ax = b, independence and dimension, and the big picture of the four subspaces. 7 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.
A subspace is a set you cannot leave by adding and scaling: a line or plane through the origin. Examples, non-examples and live Python. Strang §3.1.
02 · Strang §3.2 · The nullspace of AThe null space is every input x with Ax = 0: inputs that produce no output. A two-thruster boat, a worked example and an interactive plot in Python. Strang §3.2.
03 · Strang §3.2 · Solving Ax = 0 and Rx = 0Reduce A to R, spot the pivot and free columns, and build the special solutions that fill the null space. Worked 2×4 example with Python. Strang §3.2.
04 · Strang §3.3 · The complete solution to Ax = bEvery solution of Ax = b is one particular solution plus any null space vector. See the solution line in an interactive plot and find it in Python. Strang §3.3.
05 · Strang §3.4 · Independence, basis and dimensionIndependent vectors, a basis and the dimension of a space, with a toolkit analogy, a 3D example and rank in Python. Strang §3.4.
06 · Strang §3.5 · Dimensions of the four subspacesRow space, null space, column space and left null space: where they live, their dimensions, and why the row space is perpendicular to the null space. Interactive plot. Strang §3.5.
07 · Recap · ReadingChapter 3 on one page: subspaces, the null space, pivot and free columns, the complete solution to Ax = b, basis and dimension, and the four fundamental subspaces.
✎ · PracticeWorked examples and practice problems with hidden solutions: subspaces, null space, special solutions, the complete solution to Ax = b, independence and the four subspaces.