# Orthogonal complement question

1. Apr 24, 2006

### Geekster

I have the set

$$s = span ( [[0][1][-1][1]]^{T} )$$

And I need to find the orthogonal complement of the set.

It seems like it should be straight foward, but I'm a bit confused. I know that S is a subspace of R^4, and that there should be three free vairables.

What I did so far is to take the column vector given, and I need to find the null space of its transpose. The three free variables I picked are $$x_1= s, x_2=t, x_3=w, x_4=t-w$$.

However, x_1=s is throwing me off because its always zero. I guess what I'm really asking is, what exactly is the solution space of the homogenous system,
$$Ax=0$$
in this problem?

Thanks

Last edited: Apr 24, 2006
2. Apr 24, 2006

### 0rthodontist

You row reduce it (actually it is already row reduced) and you get 3 free variables: x1, x3, and x4. You have a pivot for x2. Your equations will be
x1 = x1
x2 = (an expression involving some of x1, x2, x3)
x3 = x3
x4 = x4