Find an orthonormal basis of the hyperplane $x_1 + x_2 + x_3 + x_4 + x_5 = 0$.
So I understand how to use Gram-Schmidt to solve this, but I'm having issues finding the basis to start with. Is it as follows:
$x_1 = -x_2 - x_3 - x_4 - x_5$, therefore our basis is:
${\begin{bmatrix}1 \\ -1 \\ 0 \\ 0 \\ 0 \end{bmatrix}},{\begin{bmatrix}1 \\ 0 \\ -1 \\ 0 \\ 0 \end{bmatrix}},{\begin{bmatrix}1 \\ 0 \\ 0 \\ -1 \\ 0 \end{bmatrix}},{\begin{bmatrix}1 \\ 0 \\ 0 \\ 0 \\ -1 \end{bmatrix}}$
And from there, we do G-S?
Yes they satisfy the equation, are 4 and are clearly linearly independent thus they span the hyperplane.
Yes to get an orthonormal basis you need Gram-Schmidt now. Let obtain a orthonormal basis before by GS and then normalize all the vectors only at the end of the process. It will simplify a lot the calculation avoiding square roots.
Are you aware about how Gram-Schmidt works?
Note that to simplify further you could choose
$${\begin{bmatrix}1 \\ -1 \\ 0 \\ 0 \\ 0 \end{bmatrix}},{\begin{bmatrix}1 \\ 1 \\ -1 \\ -1 \\ 0 \end{bmatrix}},{\begin{bmatrix}0 \\ 0 \\ 1 \\ -1 \\ 0 \end{bmatrix}},{\begin{bmatrix}1 \\ 0 \\ 0 \\ 0 \\ -1 \end{bmatrix}}$$
or similar in such way to satisfy all the conditions with the advantage to have some of them orthogonal.