Homework Help: Function that is an isomorphism

1. Jan 1, 2014

NATURE.M

1. The problem statement, all variables and given/known data

So my text states the proposition:
If V and W are finite dimensional vector spaces, then there is an isomorphism T:V→W ⇔ dim(V)=dim(W).

So, in an example the text give the transformation T:P$_{3}$(R)→P$_{3}$(R)
defined by T(p(x)) = x dp(x)/dx.

Now I understand T is not an isomorphism since ker(T) = span(1) , the set of all constant polynomial functions. But by the above theorem since the dim(V) = dim(W), T would an isomorphism.
So I'm a bit confused.

2. Jan 1, 2014

R136a1

The theorem says that there is some function that is an isomorphism. It does not say that any function $T$ is an isomorphism.

In your example, the theorem is satisfied since $S(x) = x$ is an isomorphism. But the theorem does not say that any arbitrary $T$ is an isomorphism. So you can not deduce that your $T$ is an isomorphism from the theorem.

3. Jan 1, 2014

LCKurtz

No. The theorem says that if dim(V) = dim(W) then there is an isomorphism. It doesn't say any old map you choose is an isomorphism. Your map isn't onto.

4. Jan 1, 2014

NATURE.M

Ok thanks alot. Its makes sense now.