Let V be a finite-dimensional vector space over F, and let $\displaystyle B=(x_1 , ...x_{n}) $ be an ordered basis for V. Let Q be an n by n invertible matrix with entries from F.

Define $\displaystyle x'_{j}= \sum_{i=1}^{n} Q_{ij}x_{i}$, where $\displaystyle 1 \le j \le n$

and set $\displaystyle B'=(x'_{1}, ...x'_{n})$.

Prove that B' is a basis for V and that Q in the change of coordinate matrix from B to B'.

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I've been thinking about this for a while. For B' to be a basis for V, it must be linearly independent and span(V). I just can't show it's linearly independent for the life of me. I've tried thinking of several ways to represent the relation of B' to QB' but nothing is working well. Should I make matrix representations of B and B'?

I need a little push.