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 :)