## Linear Algebra- Quadratic form and change of basis

1. The problem statement, all variables and given/known data

Suppose that for each v = (x1, x1, ... xn) in Rn, q(v) = XTAX for the given matrix A. For the given basis B of Rn, find the expression for q(v) in terms of the coordiantes yi of v relative to B.

a) A = $$\begin{bmatrix}3/2&{\sqrt{2}}&-1/2\\{\sqrt{2}}&1&-{\sqrt{2}}\\-1/2&{\sqrt{2}}&-5/2\end{bmatrix} B = {(1,0,1), (3, {\sqrt{2}}, 1), (3{\sqrt{2}}, -4, {\sqrt{2}})$$

2. Relevant equations

3. The attempt at a solution

So I read the theorem about A wrt to B is PTAP where P is the change of basis matrix from B to E3.. so I can just get A wrt to B by doing that right? I transpose P and the multiply it out like that? And it should give me a diagonal matrix?

But when I do that I get something really wrong? Like nothing is diagonal.
Is it possible for the result to have a zero column and q(v) wrt to B will not have all the terms?

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