# Matrix proof involving invertibility and bases

• Mar 23rd 2010, 03:59 PM
crymorenoobs
Matrix proof involving invertibility and bases
Ok, so I need to do the following:

Let A be an invertible n by n matrix and let {v1, ..., vn} be a basis for R^n. Prove that {Av1, ..., Avn} is also a basis for R^n.

Any ideas on how I could go about doing this??
• Mar 23rd 2010, 07:25 PM
tonio
Quote:

Originally Posted by crymorenoobs
Ok, so I need to do the following:

Let A be an invertible n by n matrix and let {v1, ..., vn} be a basis for R^n. Prove that {Av1, ..., Avn} is also a basis for R^n.

Any ideas on how I could go about doing this??

You only need to show that $\{Av_1,\ldots,Av_n\}$ is lin. indep.:

For $a_i\in\mathbb{F}=$ the definition field , $0=\sum^n_{i=1}a_iAv_i=\sum^n_{i=1}A(a_iv_i)=A\left (\sum^n_{i=1}a_iv_i\right)$ . Now, since $A$ is invertible we have that $\ker(A)=Null(A)=\{0\}$ , and then use that $\{v_1,\ldots,v_\}$ is a basis ...

Tonio
• Mar 23rd 2010, 08:14 PM
crymorenoobs
Oh ok, thanks so much!