Linear Algebra Transition Matrix Proof

[lkyabber]

Homework Statement

Prove the following theorem:
Suppose that B, C, and D are ordered bases for a nontrivial finite dimensional vector space V. let P be the transition matrix from B to C, and let Q be the transition matrix from C to D. Then QP is the transition matrix from B to D.

Homework Equations

For every v contained in V: P[v]_B=[v]_c
For every v contained in V: Q[v]_c=[v]_D

3. The Attempt at a Solution

Not sure how to go about doing this proof..don't even know where to start. Any help is greatly appreciated.

 

[D.W.]

Write out what you think QP could look like

 

[lkyabber]

Would it be the ith column of [b_i]_D?
But I'm still not sure what that looks like