Let $V$ be a vector space over $F(=\mathbb{R} ~\text{or}~\mathbb{C}), $ and let $W$ be an inner product space over $F$ with inner product $\langle \cdot, \cdot \rangle $. If $T:V\to W$ is linear, prove that $\langle x, y \rangle = \langle T(x), T(y)\rangle$ defines an inner product on $V$ iff $T$ is one-to-one.
I have tried to test all the axioms for inner product space but that does not help me to solve this problem. This seems not very tough but I am unable to hit at the right idea. So please pardon my inability to show my work as there is none to write down. Please help me to solve this. Any idea, hints will be helpful. Thanks.