I didn’t do as well as I’d have liked to on today’s Linear Algebra exam. There were 19 problems (originally 20, but the professor dropped one because it was on material we haven’t yet covered). I’m pretty sure I missed at least 3 and part of a fourth, so I’ll be lucky to get a B on this exam. I suppose I shouldn’t worry about it too much, but I really want to get an A for the course.
The final problem was a proof, which if I recall correctly was:
Given a matrix A whose columns form an orthonormal basis for a vector space, prove that AT = A-1.
The instructor gave the hint that we could do this by proving that ATA = I.
I remembered that R(A), the column space of A, is orthogonal to N(AT), the nullspace of the transpose of A, and that the direct sum of R(A) and N(AT) is Rn. But I couldn’t figure out where to take it from there.
People got all bent out of shape when Talking Barbie said “math is hard”. Did those people ever take math classes requiring proofs?
Update: I got an A (just barely) because I got partial credit for a problem I thought I’d completely missed, and Professor Sekhon decided to make the proof an extra-credit problem.
The proof turns out to be quite simple. Each element Pij of the product ATA is the dot product of two columns of A. Since the columns are orthonormal, Pij is 1 if i=j and 0 if i≠j, since the dot product of a normal vector with itself must be one, and the dot product of a vector with an orthogonal vector is 0. Too bad I couldn’t think of that yesterday.