Think about the way matrices act inside inner products. You know that
Does this give you any ideas?
FIRST LET ME SAY: I would like to be able to do this problem for myself, so if you want to actually do the problem, please don't post the answer right away so I can try it myself first.
The problem is: suppose the vector x =
and A is a (3x3) matrix such that (Atranspose)(A)(x) =
what is || Ax || ^2
So far I'm pretty much completely lost. I was thinking I could set (Atranspose)(A) equal to C and then solve for Cx=[2,2,0] but to be honest I wasn't sure even how to do that. I know that if x had been a square matrix I could have multiplied both sides by the inverse of x but obviously x isn't a square matrix.
Even using the fact that is symmetric won't give you enough equations to solve for the elements of .
You can't even write the equation Bx = C, if both B and C are matrices. The dimensions don't agree. The LHS is a vector, the RHS is a matrix.How would I go about solving Bx = C if I know what x and C are but x isn't square?
Look at this:
What does that tell you?
I'm a little confused with your use of
What exactly does that represent? I thought at first you were writing (x, ) but that doesn't make sense. Perhaps something I haven't learned yet?
Looking at what else you have written though, if then I could multiply both sides of my original equation by and get an answer of: 
Am I on the right train of thought here? Also, I haven't ever seen a theorem that would help me prove that Is there an easy way to prove this?
It's the inner product. In this case, inner product = dot product. I like the angled brackets, because it enables me to distinguish from ordered pairs, or intervals. Parentheses are used for so many darn things!
So, I'm using the notation where the last multiplication is matrix multiplication.
oooooh, Ok I think I got it now.
So for this problem all I would have to do is show that and with that, I would get that . Correct?
edit: oh and that [2,2,0] matrix should be the (3x1) matrix I had before. I didn't know how to make it a 3x1 and not a 1x3