# Dual space of the vector space

Could anyone give me a hint please :

Definition 1 Let $V$ be a vector space over the field $F$. The set $$V^* = \mathcal{L}(V,F)= \{T : V \rightarrow E| T \ \mbox{is a linear trasnformation}\}$$ is called the dual space of $V$.

Definition 2 Let $V$ be a vector space over a field $F$. For any subset $S$ of $V$, define $$S^o = \{f \in V^*| f(x) = 0 \ \forall x \in S\}.$$

$\textbf{Let$V$be a finite-dimensional vector space and$U, W$is sunspace of$V$. If$V = U \oplus W$, then$V^* = U^o \oplus W^o.$}$

Definition 3 Let $T : V \rightarrow W$ be a linear map. Define $T^t : W^* \rightarrow V^*$ by $$T^t(f) = f \circ T$$ for any $f \in W^*.$

$\textbf{Show that$T$is$1-1 $iff$T^t$is onto and$T$is onto iff$T^t$is$1-1$. Furthermore, if$T : V \rightarrow W$is a linear map where$V$is finite-dimensional, rank$T$= rank$T^t$.}$

For the first question, for, $f\in V^*$ define $f_U (u\oplus w)=f (u)$ and $f_W (u\oplus w)=f (w)$. Then $f=f_U\oplus f_W$.
For the second question: suppose $T^t$ is onto, so any $g\in V^*$ is of the form $g=f\circ T$; if $Tx=Ty$, then $g (x)=f\circ T (x)=f\circ T (y)=g (y)$. But this implies $x=y$, since the dual separates points. So $T$ is one-to-one. With this same ide you can prove the converse.