No. These two are isomorphic does not show that the map T is an isomorphism. However, since these two spaces have the same dimension, it suffices to prove that the kernel (or nullspace) of T is trivial, which is easy (but not so trivial) to prove.
Let be distinct real numbers.
Let the linear map be defined by for all in
Prove that is an isomorphism
For this question is it enough to say that and are both of dimension and therefore isomorphic? Im not sure i can apply this to a transformation. Thanks for any help!
Well its clear that the only polynomial that gives 0 for all values of x is the 0 polynomial therefore the kernal of T is trivial. Im confused as to how knowing that the kernal of T consist of only the 0 polynomial. And since the kernal of T is 0, it is therefore invertible. Which makes it isomorphic? Is this correct?