What counts as a space
A vector space is any collection of vectors closed under the two operations: add two of them and you stay inside; scale one and you stay inside. A subspace is a space living inside a bigger one.
A useful test in 2D/3D: a line or plane is a subspace only if it passes through the origin. Shift it away from the origin and scaling a vector leaves the set — so it is not a subspace. Every subspace must contain the zero vector.
The column space: which b are reachable
The column space of A is the span of its columns — every vector b you can produce as Ax. This is the payoff of the column picture: Ax = b has a solution exactly when b lies in the column space. If b is outside it, no x will ever reach it.
The nullspace: what gets sent to zero
The nullspace is every vector x with Ax = 0. It always contains the zero vector; the question is whether it contains anything else. A non-trivial nullspace means the columns are dependent, which means A is not invertible — the same fault line again, now phrased as "several inputs collapse to zero".
Rank, and the rank–nullity theorem
The rank of A is the number of pivots — equal to the number of independent columns and to the dimension of the column space. It measures how much of the output space A can actually reach.
For an m × n matrix, the counting law is exact: rank + nullity = n (the number of columns). Every column either contributes a new independent direction to the output (a pivot) or a free variable to the nullspace — and the two counts always add to n.
The four fundamental subspaces
A matrix comes with four subspaces: the column space and nullspace of A, plus the column space and nullspace of Aᵀ (the row space and left nullspace). Strang built much of introductory linear algebra around how these four fit together — they are the anatomy of a matrix.
A worked example
For A = [[1, 2], [2, 4]], find the rank, the column space, and the nullspace.
- Row-reduce: R2 − 2·R1 gives [[1, 2], [0, 0]] — one pivot. Rank = 1.The second column is a multiple of the first, so only one independent direction.
- Column space = all multiples of [1, 2] — a line through the origin.Both columns lie on that line, so their span is the line.
- Nullspace: solve x + 2y = 0 → x = −2y, giving multiples of [−2, 1].These are the vectors A sends to zero.
- Check rank + nullity = 1 + 1 = 2 = number of columns.The rank–nullity theorem holds.
Rank 1; column space is the line through [1, 2]; nullspace is the line through [−2, 1]; and 1 + 1 = 2 confirms rank–nullity.