I got a subspace $W=[(1,-3,0,1),(1,5,2,3),(0,4,1,1),(1,-2,0,4)]$
and I want to make an orthogonal projection of a vector $p=(1,0,0,0)$
onto $W$ and onto the orhhogonal complement of $W$.
Well, $W$ is $[w_1,w_2,w_3]=[(1,-3,0,1),(0,4,1,1),(1,-2,0,4)]$ for example and the orthogonal complement of $W$ is $W' = [(-10,-3,11,1)]$
So, we can project $p$ onto $W$ by the formula:
$$p_{\| W} = \sum_{i =1}^k \frac{\langle p, w_i\rangle}{\langle w_i, w_i \rangle} w_i = \frac{\langle p , w_1 \rangle}{\langle w_1 , w_1 \rangle} w_1 + \dots + \frac{\langle p , w_k \rangle}{\langle w_k , w_k \rangle} w_k.$$
But I am somehow confused. Do the vectors $w_i$ have to be orthogonal to each other before I do the projection with the vector $p$? Because $W$ is not orthogonal itself. And examples I found contained the orthogonal subspaces already (so no Gramm-Schmidt process was needed).
Best Answer
It is possible to project without an orthonormal basis.
Our space of interest $W$ is the column space of the matrix $A$ given by $$ A=\left[\begin{array}{rrrr} 1 & 1 & 0 & 1 \\ -3 & 5 & 4 & -2 \\ 0 & 2 & 1 & 0 \\ 1 & 3 & 1 & 4 \end{array}\right] $$ A basis of $W=\operatorname{Col}(A)$ is given by the columns of $A$ corresponding to pivot columns in $\operatorname{rref}(A)$. The reduced row echelon form of $A$ is $$ \operatorname{rref}(A)=\left[\begin{array}{rrrr} 1 & 0 & -\frac{1}{2} & 0 \\ 0 & 1 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] $$ This shows that $A$ has rank three and that the vectors \begin{align*} \left\langle1,\,-3,\,0,\,1\right\rangle && \left\langle1,\,5,\,2,\,3\right\rangle && \left\langle1,\,-2,\,0,\,4\right\rangle \end{align*} form a basis of $W$.
Now, put these basis vectors into the columns of a matrix $$ X = \left[\begin{array}{rrr} 1 & 1 & 1 \\ -3 & 5 & -2 \\ 0 & 2 & 0 \\ 1 & 3 & 4 \end{array}\right] $$ The projection matrix onto $W$ is $$ P=X(X^\top X)^{-1}X^\top $$ In our case, we end up with $$ P=\left[\begin{array}{rrrr} \frac{131}{231} & -\frac{10}{77} & \frac{10}{21} & \frac{10}{231} \\ -\frac{10}{77} & \frac{74}{77} & \frac{1}{7} & \frac{1}{77} \\ \frac{10}{21} & \frac{1}{7} & \frac{10}{21} & -\frac{1}{21} \\ \frac{10}{231} & \frac{1}{77} & -\frac{1}{21} & \frac{230}{231} \end{array}\right] $$ We can now project any vector $v\in\Bbb R^4$ onto $W$ by computing $Pv$.
If you're dead set on using orthonormal bases, then you could take these three basis vectors and apply the Gram-Schmidt algorithm. This gives the new basis \begin{align*} q_1 &= \left\langle\frac{1}{\sqrt{11}},\,-\frac{3}{\sqrt{11}},\,0,\,\frac{1}{\sqrt{11}}\right\rangle & q_2 &= \left\langle\frac{1}{\sqrt{7}},\,\frac{1}{\sqrt{7}},\,\frac{1}{\sqrt{7}},\,\frac{2}{\sqrt{7}}\right\rangle & q_3 &= \left\langle-\frac{1}{\sqrt{3}},\,0,\,-\frac{1}{\sqrt{3}},\,\frac{1}{\sqrt{3}}\right\rangle \end{align*} We can then project any $v\in\Bbb R^4$ with the formula $$ Pv=\langle q_1, v\rangle q_1 + \langle q_2, v\rangle q_2 + \langle q_3, v\rangle q_3 $$