Let $M$ be a subspace of $\mathbb R^4$ which is spanned by the vectors $v_1 = (1,0,-1,1)$ , $v_2=(0,1,2,1)$. Find the orthogonal complement $M^T$ of $M$ and the orthogonal projections of the vector $v=(4,3,0,1)$ on the subspaces $M,M^T$.
My solution for the first part :
Let $A=[v_1 | v_2]$, $v_1,v_2$ are already in an opened-form and linearly independent and thus solving : $A\vec x=0$ yields : $\{x - z + w = 0, y + 2z + w = 0\}$ which gives you two linearly independent vectors : $v_3,v_4$ : $v_3 = (1,-2,0,1)$,$v_4=(0,-3,1,1)$ and we get that : $M^T = span(v_3,v_4) = \langle v_3,v_4\rangle=\langle (1,-2,0,1),(0,-3,1,1) \rangle.$
Now, I have trouble finding (I do not know in that case) how to work for the orthogonal projection for the second part. Does it have to do with the Gram-Schmidt procedure ? I would appreciate a hint or an explanation on how to work over finding the orthogonal projections generally or in that example. Thanks in advance.