I have this problem, I think I've solved the first part, anyway, I'd like you to see it, because I'm not sure if I've proceeded right.

The problem says: Let $\displaystyle X=\mathbb{R}^2$. Find $\displaystyle M^{\perp{}}$ if:

a) $\displaystyle M=\{x\}$, where $\displaystyle x=(\xi_1,\xi_2)\neq 0$

b) A linearly independent set $\displaystyle \{x_1,x_2\}\subset{M}$

So, basically what I did in a) was stating: $\displaystyle M^{\perp{}}=\{z\in{X}|<x,y>=0\forall{y\in{Y}}\}$

Then $\displaystyle y=(\eta_1,\eta_2)$

Therefore $\displaystyle <x,y>=0 \rightarrow \xi_1 \eta_1+\xi_2 \eta_2=0 \rightarrow \frac{\xi_1}{\xi_2}=\frac{-\eta_2}{\eta_1}$

And then $\displaystyle y=(\xi_2,-\xi_1)$.

Is this right in the first place?

And for b) I've stated $\displaystyle M=\{(x,y)|x=\alpha_1 x+\alpha_2 y \neq 0 \forall \alpha_1,\alpha_2\in \mathbb{R} \}$, here x and y are vectors in X.

I know there's no another possible linearly independent vector in X, so there are no others orthogonal complements I think. But I don't know how to proceed from here.

Bye, and thank you for your help, which is always useful.