Prove linear transformation is monomorphism

Question

I have troubles with following exercise:

$(X, \langle \cdot, \cdot \rangle_X)$ and $(Y, \langle \cdot, \cdot \rangle_Y)$ are euclidean spaces. Linear transformation $f \in L(X, Y)$ has property $$\forall_{v\ \in\ X}\ \ \ \ \langle v, v \rangle_X = \langle f(v), f(v) \rangle_Y$$

a) Prove $f$ is monomorphism

b) Prove, that in fact $$\forall_{v, w\ \in\ X}\ \ \ \ \langle v, w \rangle_X = \langle f(v), f(w) \rangle_Y$$

I do not know where to start. To prove $f$ is monomorphism I could show $ker f = \{0\}$, but I cannot deduce anything helpful about $f$ with the given property. Can anyone help me?

Answer 1

Let $v \in \textbf{ker} \,f$. Then $\langle v, v \rangle = \langle fv, fv \rangle = 0$. But $\langle \cdot, \cdot \rangle$ is n inner product, meaning $\langle v, v \rangle = 0 \implies v = 0$.

Now for $b)$, consider $\langle v + w, v + w \rangle$