and...matrixes to a power

I’m working on the practice test and there are just two things that I need to clear up before I feel confident to take the test. I’m sorry for this email being so late at night, but I can also drop in on your office hours tomorrow before the test if that works better for you.

On the practice test question 2 part c I wanted to know if there was an easier way to prove that the inverse of B had the same eigenvectors as B and the eigenvalues are the inverse of B’s eigenvalues besides the time consuming way to invert the matrix the recalculate the eigenvector and eigenvalues. 

Now that the matrix is invertible, note that x^(-1) is a power of x and we talked about how to easily compute the power of a matrix using the diagonalization? 

Or would it not matter at all since the matrix was singular and cannot be inverted. And for part d it sounds like you’re asking for the linear combination, but does the phrase “expand the vector v in terms of the normalized eigenvectors of B” change that?

Expand the vector in terms of the normalized eigenvectors means exactly asking for the coefficients needed to make the linear combination

I also think that the third question is asking for the linear combination of both basis’, but if it is then I don’t know how I would use the answer for a to solve b since you could just solve it twice. 

for part b) I'm asking you to go from the linear combination in terms of one basis into the linear combination in terms of the other basis, remember how we did this in class? 

I’m mostly concerned that I don’t recognize the terminology for most of the test and I can’t find a place that lists all the ways to say them. Which is especially concerning because this is the third time I found the linear combination while working on the practice test and none of them are worded the same way, and I just want to make sure I’m doing this right before I take the test.

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