Consider the linear transformation T: V (+) W ---> V given by T(v,w) = v. Show that ker(T) is isomorphic to W. [V (+) W is the Cartesian sum or direct sum]
What does the rank-nullity theorem now give an alternative proof of?
Consider the linear transformation T: V (+) W ---> V given by T(v,w) = v. Show that ker(T) is isomorphic to W. [V (+) W is the Cartesian sum or direct sum]
What does the rank-nullity theorem now give an alternative proof of?
if $\displaystyle x=(x_1,x_2) \in Ker(T) $ means $\displaystyle x_1 = 0_{v} $, so Ker(T) = $\displaystyle (0_{v}, w) w \in W $
Since Dim($\displaystyle V \oplus W $) = Dim(V) + Dim(W)
Rank(T) = Dim(V)
so Rank(T) + Dim(Ker(T)) = Dim(V) + Dim(W)
Dim(Ker(T)) = Dim(W)
All finite dimensional vector spaces of equal dim are isomorphic.
additionally, you can set up a linear map from Ker(T) (since its a subspace of your vector space),
$\displaystyle G:Ker(T) \to W $ $\displaystyle (v_1,w_1) \to w_1 $
Its easy to see this is both a linear map and a bijection, thus they are isomorphic.