Hello,

the task is following. Let's assume that $\displaystyle S=\{y_1,y_2,y_3\}$ is orthogonal vector set of subspace $\displaystyle V$ and $\displaystyle \vec{0} \notin S$. Also $\displaystyle x \in V$. It's defined that:

$\displaystyle y=x-\sum_{i=1}^{3}\frac{(x, y_i)}{(y_i, y_i)}y_i$

So, the problem is to show that $\displaystyle y$ is orthogonal to all vectors of $\displaystyle S$.

When the sum is opened, we get:

$\displaystyle y=x-\frac{(x, y_1)}{(y_1, y_1)}y_1-\frac{(x, y_2)}{(y_2, y_2)}y_2-\frac{(x, y_3)}{(y_3, y_3)}y_3$

It seems like formula from Gram-Schmidt process, so in my opinion it must be $\displaystyle y \perp x, \forall x \in S$, because G-S process orthogonalizes vectors. I think I need a little bit more evidence to show the proposition, but I don't know how to continue. Any help is welcome. Thanks!