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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 $T^t$.} $

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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.

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