# Linear space and Dual space

I'm facing with a problem from my H.W. assignment. The main subjects in that assignment are linear functionals and dual space.

Let $V$ be a vector space, and $W\subseteq V$ be a subspace of $V$.

We define $W^0 = \{ \phi \in V^* \mid \phi(w) = 0 \; \forall w \in W\} \subseteq V^*$. Then $W^0$ is subspace of $V^*$.

Find the basis of $W^0$ in the following case:

$V = \mathbb{R}^3$ over $\mathbb{R}$, $W = \operatorname{span}\{ (1,0,1), (1,0,0) \}$.

If I wasn't clear enough in the question, tell me and I will fix it :)

I'm not looking for a detailed solution, explanation will be more than enough.

Thanks!

• In $\mathbb{R}^n$ or $\mathbb{C}^n$, you can represent all linear functionals in terms of the dot product with a vector. (This is a special case of the Riesz representation theorem from Hilbert space theory, though in finite dimensions there is an easier proof.) So you are essentially looking for all vectors which are perpendicular to $(1,0,1)$ and $(1,0,0)$. This should be a familiar problem to you at this point. – Ian Mar 15 '16 at 21:02

A generic element of $V^*$ can be written uniquely as $\phi = ae^1 + be^2 + ce^3$ for scalars $a,b,c$, where $e^i$ is the dual of the $i$th standard basis vector. Then $\phi \in W^0$ iff $0 = \phi(1,0,1) = a + c$ and $0 = \phi(1,0,0) = a$. In other words, $\phi \in W^0$ iff $a = c = 0$. Thus $W^0 = \operatorname{span}(e^2)$.