By means of Gram-Schmidt orthonormalization, find an orthonormal basis in 

$$S=\{v=\begin{pmatrix}x\_{1}\\x\_{2}\\x\_{3}\\x\_{4}\end{pmatrix}: x\_{1}-x\_{2}+2x\_{3}-3x\_{4}=0\}$$

subspace of $\left(\mathbb{R}^{4},\langle\,,\,\rangle\right)$.

What I do: 

$x\_{2}=\alpha, x\_{3}=\beta, x\_{4}=\gamma$ so $x\_{1}=\alpha-2\beta+3\gamma$ and :

$$\left(\alpha-2\beta+3\gamma,\alpha,\beta,\gamma\right)=\left(\alpha, \alpha,0,0\right)+\left(-2\beta,0, \beta,0\right)+\left(3\gamma,0,0,\gamma\right).$$

So a base is : $\{(1,1,0,0),(-2,0,1,0),(3,0,0,1)\}$ and I have to apply Gram-Schmidt for this base? 

Another question, what is the $\dim(S)$? $3$ or $4$ ? 

because I can find canonical base in $\mathbb{R}^{4}$ to write $\left(\alpha-2\beta+3\gamma,\alpha,\beta,\gamma\right)$.

Thanks :)