Column Space and Null Space

Linear Algebra (Matrices) · Axiom Academy

LESSON Column Space & Null Space Every matrix A hides two subspaces: the outputs it can reach, and the inputs it crushes to zero. 1. The Column Space — Everything You Can Reach Watch the two columns and of A appear in the output plane, then let every scalar combination of them sweep out. The region they fill is exactly the set of outputs you can produce. 2. The Null Space — Everything Crushed to Zero Now look at the input plane on the left. Vectors along the line fly through A and land on the single point in the output plane on the right — they get collapsed. Those inputs form the null space. Two subspaces, two questions: what A can reach, and what A destroys. They live in different spaces, and their sizes are the rank and the number of free variables. Scroll up to replay either subspace.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.