Suppose $\langle \cdot, \cdot \rangle_1$ and $\langle \cdot, \cdot \rangle_2$ are inner products on $V$ such that $\langle v, w \rangle_1 = 0$ if and only if $\langle v, w \rangle_2 = 0$. Prove that there is a positive number $c$ such that $\langle v, w \rangle_1 = c\langle v, w \rangle_2$ for every $v, w \in V$.

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Exercise 6.B.11

Suppose ${⟨⋅,⋅⟩}_{1}$ and ${⟨⋅,⋅⟩}_{2}$ are inner products on $V$ such that ${⟨ v, w ⟩}_{1} = 0$ if and only if ${⟨ v, w ⟩}_{2} = 0$ . Prove that there is a positive number $c$ such that ${⟨ v, w ⟩}_{1} = c {⟨ v, w ⟩}_{2}$ for every $v, w \in V$ .

Exercise 6.B.11

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