Transition Matrix and Ordered Bases

[LosTacos]

Homework Statement

Let B and C be ordered bases for ℝn. Let P be the matrix whose columns are the vectors in B and let Q be the matrix whose columns are the vectors in C. Prove that the transition matrix from B to C equals Q-1P.

Homework Equations

An ordered basis for a vector space V is an ordered n-tuple of vectors (v1,...,vn) such that the set (v1,...,vn) is a basis for V.

The Attempt at a Solution

I know that if B is the standard basis in ℝn, then the transition matrix from B to C is given by [1st vector in C 2nd vector in C ... nth vector in C]-1.

Also, if C is a standard basis in ℝn, then the transition matrix from B to C is given by [1st vector in B 2 vector in B ... nth vector in B].

Since I konw what the transition matrix is from B to C given different standard bases, I am having a difficult time relating this to teh columns of each.

Homework Equations

The Attempt at a Solution

 

[lippyka]

Let B = {b1, b2, ... , bn} and C = {c1, c2, ... , cn} be ordered bases for ℝn. Let P be the matrix whose columns are the vectors in B and let Q be the matrix whose columns are the vectors in C. We then have that P = [b1 b2 ... bn] and Q = [c1 c2 ... cn]. The transition matrix from B to C is then given by the matrix multiplication of Q-1P. That is, Q-1P = [c1 c2 ... cn] -1 [b1 b2 ... bn] = [c1*b1 + c2*b2 + ... + cn*bn c1*b2 + c2*b2 + ... + cn*bn ... c1*bn + c2*bn + ... + cn*bn] = [c1*b1 c1*b2 ... c1*bn c2*b1 c2*b2 ... c2*bn ... cn*b1 cn*b2 ... cn*bn] Therefore, the transition matrix from B to C equals Q-1P.