# Math Help - Isomorphism of kernel question

1. ## Isomorphism of kernel question

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?

2. ## Re: Isomorphism of kernel question

if $x=(x_1,x_2) \in Ker(T)$ means $x_1 = 0_{v}$, so Ker(T) = $(0_{v}, w) w \in W$
Since Dim( $V \oplus W$) = 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),
$G:Ker(T) \to W$ $(v_1,w_1) \to w_1$
Its easy to see this is both a linear map and a bijection, thus they are isomorphic.