Finding the basis that results from an inner space

Question

I'm given the inner product:

$\bigl\langle(x_1,x_2,x_3),(y_1,y_2,y_3)\bigr\rangle:=3x_1y_1+x_1y_3+y_1x_3+x_2y_2+2x_3y_3$

And I'm asked to find the orthonormal basis in respect of the above inner product that results from the normal basis $B=(e_1,e_2,e_3),$ $e_1=(1,0,0),e_2=(0,1,0), e_3=(0,0,1)$

My thought was to use the Gram-Schmidt process but use the given inner product to calculate the projections. However when I did so I ended up with the same vectors $e_1,e_2,e_3$ so the exact same basis...

I'm I doing somthing wrong or does the basis just stay the same?

Answer 1

You are doing something wrong, and you are doing it from the start. Note that $\bigl\langle(1,0,0),(1,0,0)\bigr\rangle=3$ and that therefore the first vector that you should have got was $\frac1{\sqrt3}(1,0,0)$.