Four spaces from one matrix
An m-by-n matrix A hands you four subspaces. Two live on the input side (in R^n): the row space (span of the rows) and the null space (everything A sends to zero). Two live on the output side (in R^m): the column space (span of the columns, where outputs actually land) and the left null space (the null space of the transpose A^T).
Rank-nullity ties them together
All four sizes come from one number, the rank r. Column space and row space both have dimension r — that is the theorem that row rank equals column rank wearing a new hat. On the input side, rank-nullity gives nullity = n - r. On the output side, the left null space has dimension m - r. Know r, m, and n, and you know all four.
A is m x n with rank r: row space dim = r (in R^n) null space dim = n - r (in R^n) column space dim = r (in R^m) left null space dim = m - r (in R^m) example: 3 x 4, r = 2 row=2, null=2 (sum 4 = n) col=2, left null=1 (sum 3 = m)
A preview: right angles
The four spaces are not arranged at random. On the input side, the row space and the null space meet at a perfect right angle — they are orthogonal complements. Every row of A is perpendicular to every solution of A*x = 0, which is just a restatement of A*x = 0 row by row: each row dotted with x is zero. Together they fill all of R^n with nothing left over.