A true or false about an inner product

Hi MHF,

I did badly (failed) in the final exam on linear algebra (4.4/10) and so I must retake the exam in late June or early July. I'm still stuck on some exam questions :

Tell whether this is true or false : Let and be vector spaces and an isomorphism. If has an inner product , then defines an inner product in .

I admit that I'm totally lost. I don't know what theorem(s) could be useful...

Although is an isomorfism, I don't think it imples that and have the same dimension... I don't even know what are and .

Is the identity linear transformation? If so, I guess it's almost obvious that the answer is "true".