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??
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