1. ## Linear Algebra Proof

Hi all,

I need to prove this:

If The columns of A span Rn, then The equation Ax = b has at least one solution for each b in Rn

Step by step please....i really want to understand this...thanks
thanks

2. ## Re: Linear Algebra Proof

If the columns of A span ℝn, it means that for every b ∈ ℝn there exist numbers x1, ..., xn such that x1A1 + ... + xnAn = b, where Ai is the ith column of A. But this exactly means that Ax = b has the solution x = (x1, ..., xn).

3. ## Re: Linear Algebra Proof

That's pretty much what spans means isn't it? $x_1A_1+ x_2A_2+ \cdot\cdot\cdot+ x_nA_n$ is exactly the definition of "Ax". So that is simply saying, "for any $b\in R^$, there exist x such that Ax= b".