The subspace W of $\displaystyle R_4$ is given by the vectors:

u=(1,1,0,1)

v=(1,-1,1,0)

w=(1,-1,0,-1)

I need help calculating the projection of the vector x (1,1,1,1) in to the subspace.

I know I need to make all the vectors orthogonal and then i can use the following formula:

$\displaystyle \dfrac{x*u}{\|u^2\|}u+\dfrac{x*v}{\|v^2\|}v+\dfrac {x*w}{\|w^2\|}w$

My question is can i use the first two vectors (that happen to be orthogonal) and just use Gram Schmidt on the last vector to make it orthogonal or do I have to use Gram-Schmidt on all the vectors?

I tried using Gram-schmidt on the third vector to get it orthogonal, this got me the following vector.$\displaystyle \begin{bmatrix}2/3\\0\\-2/3\\-2/3\\2\end{bmatrix}$

I put this vector in to the formula above and came up with the following projection:$\displaystyle \begin{bmatrix}1\\2/3\\2/3\\4/3\end{bmatrix}$

Are my calculations correct?

/Maria