Determining when a linear transformation is an inner product

Question

Hi,

I know that $$ \langle v,w\rangle $$ satisfies the first three conditions of being an inner product always. As such, the only condition that needs to be met is

$$ T(v) \cdot T(v) > 0 $$

This is true when $T(v)$ is non-zero. But the answer in the back of the book says this is true when the $ker(T) = 0$. Can someone explain what this means?

Thanks

Answer 1

Suppose that $\ker T \neq \{0\}$, so there is some vector $v \in \ker T$, which means $T(v) = 0$. Then $$ \langle v, v \rangle = T(v) \cdot T(v) = 0 $$ but $v \neq 0$, which contradicts positive-definiteness of inner-products.