from an orthonormal basis of polynomials, my corresponding coordinate matrix is not orthogonal?

Question

how could I do it? do I just put the coordinates of each orthonormal polynomial vector into a column, w.r.t. the standard basis?

for example, if my orthonormal basis is

$$\{1, 2x, 3x^2\}$$

Then is the first column of my orthogonal matrix just

$$[1,0,0]^t$$

the second column

$$[0,2,0]^t$$

and third column

$$[0,0,3]^t$$?

the matrix is obviously not orthogonal, just by inspection, but my basis is orthonormal for some specified inner product. so this is confusing me...

Answer 1

my basis $1,x,x^2$ is orthonormal for some specified inner product

is refuted in Is there a representation of an inner product where monomials are orthogonal?