# Thread: 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 $\displaystyle \{Av_1,\ldots,Av_n\}$ is lin. indep.:

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

Tonio

3. Oh ok, thanks so much!