Math Help - Determine change of basis matrix

The question i'm given is to find the change of basis matrix P_{C<-B . I'm given the bases for both B and C but I'm a bit clueless as to how to solve the question. I figure what they are asking is to find a matrix which turns a vector from the span B to be in the span C instead.}

Hey dipsy34.

I'll start you off with a hint: If x is the vector in the standard basis (identity matrix) then we have c = Cx b = Bx where C and B are the change of basis matrix to go from I (identity) to C or B.

Hint: Given the hint try getting c in terms of B,C,b and you will have your answer (remember that you can remove a matrix by pre-multiplying it by its inverse matrix).

chiro in your hint are c and b vectors in C respective B or basis?

Yes they are the vectors in the corresponding basis. So b is the vector in basis B and c is the vector in basis C. x is the basis vector in the normal basis R^n.

I think I got it now, I take all the vectors that make up the basis B, b₁,b₂... ...b_k and transforms them using C and the change of basis matrix P_C<-B would then consist of the transformed vectors. If they both are instandard basis that is, if not I would have to take: P_C<-B = B^-1​C?

Yes, apply the given linear transformation to each of the basis vectors for the domain space, in turn. Write each result as a linear combination of the given basis vectors for the range space. The coefficients are a column in the matrix.

(You really have to have ordered bases. The same basis vectors in a different order will give a different matrix.)