Fourier coefficients with respect to an orthonormal basis for an inner product space 
$V = \operatorname{span}(S)$, where $S = \{(1, i, 0), (1 - i, 2, 4i)\}$, and $x = (3 + i, 4i, -4)$.
  
  
*
  
*Apply the Gram–Schmidt process to the given subset $S$ of the inner
  product space $V$ to obtain an orthogonal basis for $\operatorname{span}(S)$.
  
*Normalize the vectors in this basis to obtain an orthonormal basis $B$ for $\operatorname{span}(S)$.  
  
*Compute the Fourier coefficients of the given vector relative to $0$ by solving the system of equations $a_1u_1 + a_2u_2 + a_3u_3 = x$, where the $u_j$ are the vectors of basis $B$.
  
*Compute the Fourier coefficients directly by $a_i = \langle x, u_i\rangle$.
  

I'm pretty sure I know how to, and I did the first two parts correctly; the computed orthonormal basis is:
$$B = \left\{ \frac{1}{\sqrt2}(1, i, 0),  \frac{1}{\sqrt{17}}(1 + i, 1 - i, 4i)\right\}.$$
The correct answer for the Fourier coefficients is 
$$\frac{7 + i}{\sqrt 2}, \sqrt{17} i.$$
But I can't seem to get either of them through either method. I can't get the answer through (3), but I feel it's just a calculation error. I am really not sure how to compute (4) though — can someone show me the step-by-step computation to get the Fourier coefficient through (4)?
Also, if my orthonormal basis is incorrect, please show me your answer too.

$B$ is not orthonormal, because it is not orthogonal. In order to get an orthonormal basis, you first need a orthogonal basis.
