If the columns of A span ℝ^{n}, it means that for every b ∈ ℝ^{n}there exist numbers x_{1}, ..., x_{n}such that x_{1}A_{1}+ ... + x_{n}A_{n}= b, where A_{i}is theith column of A. But this exactly means that Ax = b has the solution x = (x_{1}, ..., x_{n}).