# Math Help - Matrix proof involving invertibility and bases

1. ## 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??

2. 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

3. Oh ok, thanks so much!