Finding all vectors orthogonal to a span V

Question

I'm given u1 = (1,2,2,-1) u2 =(1,1,-1,1) u3 = (-1,1,-1,-1) V = span(u1, u2, u3) How do I find all vectors that are orthogonal to V?

Answer 1

A vector $x=(x_1,x_2,x_3,x_4)$ is orthogonal to $V$ if and only if $x$ is orthogonal to each vector in $V$. It is not hard to convince yourself that it is enough that $x$ is orthogonal to each of the $u_i$'s so that it is orthogonal to each vector in $V$.

Hence, given $$\left< x,u_i \right> = 0 \qquad (i=1,2,3)$$ you can form an homogeneous system of linear equations and solve it to find some conditions on the coordinates of $x$.

You end up with $x_1=-x_4$ and $x_2=x_3=\tfrac12 x_4$.

Therefore, all the vectors that are orthogonal to $\textsf V$ are in the form of $$(x_1,x_2,x_3,x_4) = (-t,\tfrac{t}{2},\tfrac{t}{2},t) = t(-1,\tfrac12,\tfrac12,1)$$ for some $t\in\Bbb R$.