# Thread:

1. ## Finite Dimensional Vector Space

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

2. Originally Posted by ques
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 $\geq$ instead of $\leq.$ let $T^{-1}(Z)=X$ and define the map $S: V \longrightarrow W/Z$ by $S(v)=T(v) + Z.$ clearly $\ker S = X$ and thus $V/X \cong S(V)=T(V)/Z.$

hence $\dim V - \dim X = \dim V/X = \dim T(V)/Z = \dim T(V) - \dim Z \leq \dim W - \dim Z.$ therefore $\dim X \geq \dim V - \dim W + \dim Z.$
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