Let V, W be finite dimensional vector spaces over a
field k and let Z subset of W be a
subspace. Let T : V -> W be a linear map. Prove that
dim( T^(-1) ( Z ) ) <= dim V - dim W + dim Z
your inequality is not correct. it should be $\displaystyle \geq$ instead of $\displaystyle \leq.$ let $\displaystyle T^{-1}(Z)=X$ and define the map $\displaystyle S: V \longrightarrow W/Z$ by $\displaystyle S(v)=T(v) + Z.$ clearly $\displaystyle \ker S = X$ and thus $\displaystyle V/X \cong S(V)=T(V)/Z.$
hence $\displaystyle \dim V - \dim X = \dim V/X = \dim T(V)/Z = \dim T(V) - \dim Z \leq \dim W - \dim Z.$ therefore $\displaystyle \dim X \geq \dim V - \dim W + \dim Z.$