# Finite Dimensional Vector Space

#### 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

#### NonCommAlg

MHF Hall of Honor
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 $$\displaystyle \dim X \geq \dim V - \dim W + \dim Z.$$

ques and Bruno J.