What you can now do.

• §3.1 Subspaces. The straight path past your front door: a set you can't leave by stretching and adding. It must contain 0. In the plane: the origin, a line through it, or the whole plane. The column space is one.
• §3.2 The nullspace. Inputs that do nothing: every x with Ax = 0. A nonzero one is a recipe that mixes the columns into zero, so the columns are dependent: one column is a mix of the others (§3.4 below). For an invertible A, only x = 0.
• §3.2 R and special solutions. Four friends settling two bills that must balance to zero: you choose the free variables, and the pivot variables follow. Reduce A to R (pivots 1, zeros above and below them), set one free variable to 1 and the rest to 0. The n − r special solutions span N(A). The rank r is the number of pivots.
• §3.3 The complete solution. One route to the bakery plus any stroll that brings you back: x = x_p + x_null. Find x_p from [A | b] with the free variables set to 0. A row 0 = nonzero means b is not in C(A), and there is no solution.
• §3.4 Independence, basis and dimension. Walking directions with nothing extra: independent vectors have only the all-zero combination giving 0. A basis is independent and spans the space, and every basis has the same size, the dimension. The pivot columns are a basis of C(A), so dim C(A) = r.
• §3.5 The four subspaces. The shadow: row space and nullspace among the inputs, column space and left nullspace among the outputs. Sizes r, n − r, r, m − r. The nullspace is perpendicular to the row space.

Next lesson: orthogonality and least squares (chapter 4)

Perpendicular subspaces, projections, and the best answer to Ax = b when there is no exact one. That is how you fit a line to noisy sensor data.

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