I have the following question here.
Let $W$ be the subspace of $\mathbb{R}^5$ spanned by the vectors $$\{(1,2,2,4,-1),(1,2,-1,1,1),(0,0,-1,-1,1),(0,0,1,1,0) \}$$.
a) Find an orthogonal basis for $W$.
b) Find an orthogonal basis for $W^{\perp}$.
c) Express the vector $(5,0,0,2,0)$ as a sum of $$(5,0,0,2,0)=w+w'$$
where $w\in W$ and $w' \in W^{\perp}$.
I already did parts $a$ and $b$.
For part $a$, the last vector can be written as a linear combination of the other vectors so we can't use that in the orthogonal basis. Then I used the Gram schmist process to ensure the vectors were orthogonal to each other. So I get that my basis is:
$$\beta = \{ (1,2,2,4,-1),\frac{1}{13}(10,20,-19,1,16),\frac{1}{43}\left(\frac{-3}{2},-3,5,2,\frac{21}{2}\right)\}$$.
For part $b$, I know that that we can just find the null space of the given vectors so I row reduced the following matrix:
$$\begin{bmatrix} 1 & 2 & 2 & 4 & -1\\ 1 & 2 & -1 & 1 & 1\\ 0 & 0 & -1 & -1 & 1\\ 0 & 0 & 1 & 1 & 0\end{bmatrix}\rightarrow \begin{bmatrix} 1 & 2 & 0 & 2 & 0\\ 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0\end{bmatrix}$$
So this means I can directly extract the null space from here which is $\{(-2,1,0,0,0),(-2,0,-1,1,0)\}$. After this I apply the gram schmidt process to get:
$$\{(-2,1,0,0,0),\left(\frac{-2}{5},\frac{-4}{5},-1,1,0\right) \}$$.
However, I am not sure about part $c$. Do I first have to project $(5,0,0,2,0)$ onto each of the basis vectors for the parallel portion (AKA onto $W$) and then add the total result? Finally would I then subtract the result of adding from $(5,0,0,2,0)$ to get the perpendicular portion?
If someone could clarify, that would be great!