# proof using invertible and spanning

• Apr 25th 2013, 04:12 PM
widenerl194
proof using invertible and spanning
Let A be an n x n matrix. Denote it's columns b v1, v2, ... vn. Keep in mind these are column vectors in Rn.

Prove that A is invertible if and only if {v1, v2, ... vn} spans Rn.
• Apr 25th 2013, 05:24 PM
Gusbob
Re: proof using invertible and spanning
I have no idea what you know and can use. But here is a quick proof using things I feel ought to be done before proving this theorem.

$\displaystyle A$ spans $\displaystyle \mathbb{R}^n$

iff

$\displaystyle Ax=b$ is consistent for all $\displaystyle b \in \mathbb{R}^n$

iff

$\displaystyle A$ has $\displaystyle n$ pivot rows (and also $\displaystyle n$ pivot columns since $\displaystyle A$ is square)

iff

$\displaystyle A$ is row equivalent to the $\displaystyle n\times n$ identity matrix

iff

$\displaystyle A$ is invertible.
• Apr 25th 2013, 05:26 PM
HallsofIvy
Re: proof using invertible and spanning
First show that if you take A times $\displaystyle \begin{bmatrix}1 \\ 0 \\ 0 \\ \cdot\cdot\cdot \\ 0\end{bmatrix}$ you get the first column of A, that if you take A times $\displaystyle \begin{bmatrix} 0 \\ 1 \\ 0 \\ \cdot\cdot\cdot \\ 0 \end{bmatrix}$ you get the second column, etc.

In other words, A maps the standard basis for Rn into the columns of A. Those vectors span the range of A.