(Short deadline) Linear Algebra

Let $V$ and $W$ be vector spaces over a field $F$, $T : V ? W$ a linear map, $X$ a subspace of  $V$ and $Y$ a subspace of $W$.

Show that:

a) $(T^{-1}(Y))^0=T^*(Y^0)$
b)$T(X) ? Y$ if and only if $T^?(Y^0) ? X^0$

Answer

a) Let $f\in T^{*}(Y^0)$. Then $f=T^*(g)$ where $g\in Y^0$. Now let $v\in T^{-1}(Y)$. Then $Tv \in Y$. So  $$g(Tv)=0 \implies f(v)=(T^*(g))(v)=g(Tv)=0$$ 
Therefore $f\in (T^{-1}(Y))^0$. So $T^{*}(Y^0) \subset (T^{-1}(Y))^0$.

Now let $f\in (T^{-1}(Y))^0$. Note that $\ker T =T^{-1}(0)\subset T^{-1}(Y)$. So $f(\ker T)=\{0\}$. Now for any $w\in T(V)$ there is $v\in V$ so that $T^{-1}(w)=v+\ker T$. So $$ f(T^{-1}(w))=\{f(v)\}$$ Let us define $g\in W^*$ by  $g(w):=f(u)$ where $u$ is some arbitrary element of $ T^{-1}(Pw)$ and $P$ is some fixed projection onto $T(V)$. Note that by the above relation $g$ is well-defined. It is easy to see that $g$ is linear because $f,P$ are linear and the set $T^{-1}(w)$ depends linearly on $w$. And for any $v\in V$ we have  $$ g(Tv)=f(v) $$ since $v\in T^{-1}(Tv)=T^{-1}(PTv)$. So $f=T^*(g)$ and thus  $T^{*}(Y^0) \supset (T^{-1}(Y))^0$.

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The following proof of the above fact also works when both $V,W$ are finite dimensional:

Now we know that $ T|_{T^{-1}(Y)} :T^{-1}(Y)\to Y$ is one-to-one and onto, so by rank-nullity theorem we have $$ \dim Y + \dim \ker T = \mathrm{rank} T|_{T^{-1}(Y)} + \dim \ker T = \dim T^{-1}(Y)$$ Note that $\ker T \subset T^{-1}(Y)$ so $\ker T|_{T^{-1}(Y)} = \ker T$. So 
$$ \dim (T^{-1}(Y))^0 =\dim V - \dim T^{-1}(Y)=\dim V - (\dim Y + \dim \ker T) \\ \;\\ = \dim V - (\dim Y + \dim V -\mathrm{rank} T)=\mathrm{rank} T-\dim Y =\mathrm{rank} T^* -\dim Y\\\;\\ =\mathrm{rank} T^* -(\dim W -\dim Y^0)=\mathrm{rank} T^* -\dim W^* +\dim Y^0\\\;\\ = -\dim \ker T^* +\dim Y^0= \dim T^{*}(Y^0)$$
Note that in the last two equalities we applied the rank-nullity theorem to the map $T^* :W^* \to V^*$ and the onto map $T^*|_{Y^0} :Y^0 \to T^*(Y^0)$. We also used the fact that the dimension of a subspace plus the dimension of its annihilator is equal to the dimension of the space, and the rank of a map and its adjoint are the same.
So we must have $T^{*}(Y^0) = (T^{-1}(Y))^0$ since the smaller subspace has the same dimension as the larger one.

b) Suppose $T(X)\subset Y$. Then $X\subset T^{-1}(Y)$. So $X^0 \supset (T^{-1}(Y))^0=T^{*}(Y^0)$. Conversely suppose $(T^{-1}(Y))^0=T^{*}(Y^0)\subset X^0$. Then $$T^{-1}(Y)\supset X\implies Y\supset T(X)$$ as desired.

The answer is accepted.