I've been working on the following problem:

"Let

be a finite-dimensional vector space over a field

, and let

be an ordered basis for

. Let

be an

invertible matrix with entries from

. Define

for

, and set

. Prove that

is a basis for

and hence that

is the change of coordinate matrix changing

-coordinates into

-coordinates."

Can someone point me in the right direction with this? Obviously I need to show that

is linearly independent and spans

. I know there are n vectors in

so it only remains to show that its elements are linearly independent, correct? I can't seem to figure it out though.