Experiments / live

The four fundamental subspaces playground

A matrix sorts its output space into what it can reach and what it can't. This one has fixed columns that are parallel, so its column space C(A) — everything Ax can reach — is a single line. Drag the target b around: whenever it lands on the line the system Ax = b is solvable, and the moment it drifts off, no input can produce it. The dashed stick is the residual — the leftover that lands in the perpendicular piece. The full story — column space, null space, rank–nullity, and why solvability means b ∈ C(A) — is in the essay.

output space ℝᵐ — drag the ink dot b

b ∈ C(A): solvable N(A) direction: (1, −2)

This playground is interactive and needs JavaScript. Here's the idea as a static picture: the column space C(A) is a line (the reachable set), ŷ is the closest point on it to the target b, and the dashed residual b − ŷ is perpendicular. When b sits on the line, the residual is zero and the system is solvable; otherwise it isn't.