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 $\displaystyle V$ and $\displaystyle W$ be vector spaces and $\displaystyle T:V\to W$ an isomorphism. If $\displaystyle W$ has an inner product $\displaystyle \langle , \rangle$, then $\displaystyle (X,Y)=\langle TX, TY \rangle$ defines an inner product in $\displaystyle V$.

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

Although $\displaystyle T$ is an isomorfism, I don't think it imples that $\displaystyle V$ and $\displaystyle W$ have the same dimension... I don't even know what are $\displaystyle X$ and $\displaystyle Y$.

Is $\displaystyle T$ the identity linear transformation? If so, I guess it's almost obvious that the answer is "true".