In $\mathbb{C}^4$ with the standard vector multiplication we set $W = span\{ (1,-i,2+i,2-i),(-i,-i,0,0),(2-2i,-2-i,5,3-2i)\}$ and the question is to find the orthogonal projection of $v=(i,i,i,i)$ onto $W$.
First using Gram-Shcmidt method i got an orthogonal base for $W$ which is :
$W = span \{ (1,-i,2+i,2-i) ,(-\frac{1}{12} - \frac{11 i}{12}, \frac{1}{12} - \frac{11 i}{12}, -\frac{1}{4} + \frac{i}{12}, -\frac{1}{12} + \frac{i}{4}), (\frac{8}{11} - i, -\frac{8}{11} + i, \frac{15}{22} - \frac{5 i}{22}, \frac{5}{22} + \frac{29 i}{22})\}$,
Now i can use the method :
$ E_W(v) = \sum \limits_{i=1}^{3} \frac{<v,u_i>}{||u_i||^2 } u_i$ and i got the answer :
$ E_W(v) =(\frac{8}{59} + \frac{65 i}{59}, -\frac{8}{59} + \frac{53 i}{59}, -\frac{11}{59} + \frac{47 i}{59}, \frac{11}{59} + \frac{ 61 i}{59}) $
but the answer in the book is $ \frac{-7+6i}{124-110i} (-10i,-10i,-9-2i,5)$
Where i got wrong ? , please reply