Originally Posted by

**AlexP** I've been working on the following problem:

"Let $\displaystyle V$ be a finite-dimensional vector space over a field $\displaystyle F$, and let $\displaystyle \beta=\{x_1, ..., x_n\}$ be an ordered basis for $\displaystyle V$. Let $\displaystyle Q$ be an $\displaystyle n\times n$ invertible matrix with entries from $\displaystyle F$. Define $\displaystyle x_j'=\displaystyle\sum_{i=1}^n Q_{ij}x_i$ for $\displaystyle 1 \le j \le n$, and set $\displaystyle \beta'=\{x_1', ..., x_n'\}$. Prove that $\displaystyle \beta'$ is a basis for $\displaystyle V$ and hence that $\displaystyle Q$ is the change of coordinate matrix changing $\displaystyle \beta'$-coordinates into $\displaystyle \beta$-coordinates."

Can someone point me in the right direction with this? Obviously I need to show that $\displaystyle \beta'$ is linearly independent and spans $\displaystyle V$. I know there are n vectors in $\displaystyle \beta'$ so it only remains to show that its elements are linearly independent, correct? I can't seem to figure it out though.