Finite Dimensional Vector Space

May 2010
24
5
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
May 2008
2,295
1,663
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 \(\displaystyle \dim X \geq \dim V - \dim W + \dim Z.\)
 
  • Like
Reactions: ques and Bruno J.