proving that change of basis matrix is invertible.

Hi guys.

Let say is the change of basis matrix from the base B to base C.

I need to prove that is invertible.

So that's what i did so far:

I defined:

Now I tried to define the matrix (and to show its columns are linear independent), but this is where it got a little complicated for me:

since and B is basis for V, every vector can be represented as a linear combination of B's vectors:

.

.

.

and then, these equation can be written as:

Now, where is this change of basis matrix?

Is it the matrix with the entries?

Am I even writing it correctly?

Thanks in advanced!