Question about dual spaces and subspaces

Question

I don't understand this question at all.. this is the first time I've encountered anything regarding dual spaces and functionals and I'm completely lost.

I know the dimension of $V^{*}$ is n, and that $U^{0}$ is a subspace of $V^{*}$, I also know that $dimU=dimU^{0}≤dimV^{*}=n$

But now I don't know what to do..

Answer 1

Let $\{e_1, \dots ,e_m\}$ be a basis of $U$ that you complement into a basis $\{e_1,\dots,e_m,e_{m+1}, \dots, e_n\}$ of $V$. Let $\{e_1^*, \dots,e_n^*\}$ be the dual basis.

$\{e_{m+1}^*, \dots ,e_n^*\}$ is a basis of $U^\circ$.

It is linearly independent as it is a subset of a basis. It spans $U^\circ$ as $e_i^*(e_j)= \delta_{ij}$ for $1 \le i \le n$ and $1 \le j \le m$ where $\delta_{ij}$ stands for the Kronecker symbol.

Answer 2

Easier: the map $$V^* \ni f \mapsto f|_U \in U^*$$is surjective (why?) with kernel $U^0$ (obvious by definition of $U^0$). Since $\dim U = \dim U^*$, the rank-nullity theorem gives $$\dim U+\dim U^0=\dim V.$$