Problem:

For a set of vectors {$\displaystyle u_1,....,u_k$} $\displaystyle \subset R^n$ show that the set of vectors $\displaystyle X \subset R^n$ that are orthogonal to each $\displaystyle u_i$ is a subspace of $\displaystyle R^n$

Solution:

Alright, I'm thinking that since the set of vectors X is orthogonal to $\displaystyle u_i$ then I can say something like

{$\displaystyle u_1x_1, u_2x_2,....,u_kx_k$} = 0

therefore,

$\displaystyle u_1x_1$ = 0 , and $\displaystyle u_2x_2 $= 0

so

$\displaystyle u_1x_1$ + $\displaystyle u_2x_2$ = 0

Am I thinking this through properly?