Prove that the column space of an m x n matrix A is a subspa

Prove that the column space of an m x n matrix A is a subspace of R^m.

Solution

The proof of this fact requires verifying the three properties of subspaces, namely:

(i) ¯ 0 ? NulA;

(ii)NulA is closed under addition;

(iii) NulA is closed under scalar multiplication.

For (i), note that A¯ 0 = ¯ 0, so ¯ 0 ? NulA.

For (ii), if u ¯ and v ¯ are in NulA, then A(¯ u + ¯ v) = Au ¯ + Av ¯ = ¯ 0 + ¯ 0 = ¯ 0, so u ¯ + ¯ v ? NulA.

For (iii), if c is a scalar and u ¯ ? NulA, then A(cu ¯) = cAu ¯ = c¯ 0 = ¯ 0, so cu ¯ ? NulA