# I need help to prove this in Linear Algebra

• Mar 19th 2013, 12:17 PM
TrystanandEmma1
I need help to prove this in Linear Algebra
If The linear transformation x Ax maps Rn onto Rn.

then
The columns of A span Rn.

I proved the other way...but in order for this proof to be complete...I need to prove this..step by step please...it is important that I do understand the proof as well....thanks in advance
• Mar 19th 2013, 01:25 PM
emakarov
Re: I need help to prove this in Linear Algebra
This is similar to the other problem. Given x = (x1, ..., xn)T, where the superscript T means transpose, i.e., this is a column vector, Ax = x1A1 + ... + xnAn where Ai is the ith column of A and this is the sum of column vectors. The fact that x ↦ Ax is onto means that for every b ∈ ℝn there exist an x such that Ax = b. But this also means that the columns of A span ℝn.
• Mar 19th 2013, 01:36 PM
jakncoke
Re: I need help to prove this in Linear Algebra
Doesn't it follow from the fact that the linear transformation is "onto" ?

Means that any vector in \$\displaystyle \mathbb{R}^{n} \$ can be gotten by doing Ax, where x is the coordinate written under some basis B. That itself means that any vector in \$\displaystyle \mathbb{R}^n\$ can be written in terms of \$\displaystyle c_1 v_1 + ... +c_n v_n \$ where \$\displaystyle v_1,..v_n\$ are the columns of \$\displaystyle A\$ and \$\displaystyle c_1,..c_n\$ is a coordinate under some basis B. thus the columns of A span \$\displaystyle \mathbb{R}^n\$