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?

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- February 20th 2013, 05:52 PMTimsBobby2Isomorphism 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? - February 20th 2013, 06:10 PMjakncokeRe: Isomorphism of kernel question
if means , so Ker(T) =

Since Dim( ) = 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),

Its easy to see this is both a linear map and a bijection, thus they are isomorphic.