# Question ​​​​​​​ Problem 1 (25 points). Let $$U$$ and $$V$$ be real vector spaces (i.e. vector spaces over $$\mathbb{R})$$, equipped with scalar products $$\langle\cdot, \cdot\rangle_{U}$$ and $$\langle\cdot, \cdot\rangle_{V}$$, respectively. Let $$f: U \rightarrow V$$ be a function that satisfies the property that $\langle f(\mathbf{u}), f(\mathbf{v})\rangle_{V}=\langle\mathbf{u}, \mathbf{v}\rangle_{U}$ for all $$\mathbf{u}, \mathbf{v} \in U$$. Prove that $$f$$ is linear and one-to-one. Hint: For linearity, consider - $$\langle f(\mathbf{u}+\mathbf{v})-f(\mathbf{u})-f(\mathbf{v}), f(\mathbf{u}+\mathbf{v})-f(\mathbf{u})-f(\mathbf{v})\rangle_{V}$$ - $$\langle f(\alpha \mathbf{u})-\alpha f(\mathbf{u}), f(\alpha \mathbf{u})-\alpha f(\mathbf{u})\rangle_{V}$$ where $$\mathbf{u}, \mathbf{v} \in U$$ and $$\alpha \in \mathbb{R}$$.

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Transcribed Image Text: Problem 1 (25 points). Let $$U$$ and $$V$$ be real vector spaces (i.e. vector spaces over $$\mathbb{R})$$, equipped with scalar products $$\langle\cdot, \cdot\rangle_{U}$$ and $$\langle\cdot, \cdot\rangle_{V}$$, respectively. Let $$f: U \rightarrow V$$ be a function that satisfies the property that $\langle f(\mathbf{u}), f(\mathbf{v})\rangle_{V}=\langle\mathbf{u}, \mathbf{v}\rangle_{U}$ for all $$\mathbf{u}, \mathbf{v} \in U$$. Prove that $$f$$ is linear and one-to-one. Hint: For linearity, consider - $$\langle f(\mathbf{u}+\mathbf{v})-f(\mathbf{u})-f(\mathbf{v}), f(\mathbf{u}+\mathbf{v})-f(\mathbf{u})-f(\mathbf{v})\rangle_{V}$$ - $$\langle f(\alpha \mathbf{u})-\alpha f(\mathbf{u}), f(\alpha \mathbf{u})-\alpha f(\mathbf{u})\rangle_{V}$$ where $$\mathbf{u}, \mathbf{v} \in U$$ and $$\alpha \in \mathbb{R}$$.
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Transcribed Image Text: Problem 1 (25 points). Let $$U$$ and $$V$$ be real vector spaces (i.e. vector spaces over $$\mathbb{R})$$, equipped with scalar products $$\langle\cdot, \cdot\rangle_{U}$$ and $$\langle\cdot, \cdot\rangle_{V}$$, respectively. Let $$f: U \rightarrow V$$ be a function that satisfies the property that $\langle f(\mathbf{u}), f(\mathbf{v})\rangle_{V}=\langle\mathbf{u}, \mathbf{v}\rangle_{U}$ for all $$\mathbf{u}, \mathbf{v} \in U$$. Prove that $$f$$ is linear and one-to-one. Hint: For linearity, consider - $$\langle f(\mathbf{u}+\mathbf{v})-f(\mathbf{u})-f(\mathbf{v}), f(\mathbf{u}+\mathbf{v})-f(\mathbf{u})-f(\mathbf{v})\rangle_{V}$$ - $$\langle f(\alpha \mathbf{u})-\alpha f(\mathbf{u}), f(\alpha \mathbf{u})-\alpha f(\mathbf{u})\rangle_{V}$$ where $$\mathbf{u}, \mathbf{v} \in U$$ and $$\alpha \in \mathbb{R}$$.
&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/3Given: &#160;LetU andV be real vector spaces (i.e. vector spaces overR), equipped with scalar products $$\mathrm{⟨⋅,⋅⟩}$$ U&#8203;and$$\mathrm{⟨⋅,⋅⟩}$$V&#8203;, respectively. Let$$\mathrm{{f}:{U}→{V}}$$be a function that satisfies the property that$$\mathrm{{\left[{\left\langle{f{{\left({\mathbf{{{u}}}}\right)}}},{f{{\left({\mathbf{{{v}}}}\right)}}}\right\rangle}_{{{V}}}={\left\langle{\mathbf{{{u}}}},{\mathbf{{{v}}}}\right\rangle}_{{{U}}}\ \right]}\ {f}{o}{r}\ {a}{l}{l}\ {u},{v}∈{U}}$$Aim: To prove that $$\mathrm{{f}}$$ is linear and one-to-one.Explanation:Please refer to solution in this step.Step2/3To prove that $$\mathrm{{f}}$$is linear and one-to-one, we will use the given property and the hint.To prove linearity, we need to show that for any$$\mathrm{{u},{v}∈{U}}$$and$$\mathrm{α∈{R}}$$, we have$$\mathrm{{f{{\left(α{u}+{v}\right)}}}=α{f{{\left({u}\right)}}}+{f{{\left({v}\right)}}}}$$Consider$$\mathrm{{w}\:={f{{\left({u}+{v}\right)}}}-{f{{\left({u}\right)}}}-{f{{\left({v}\right)}}}}$$Then, using the given property, we have$$\mathrm{⟨{w},{w}⟩}$$$$\mathrm{=⟨{f{{\left({u}+{v}\right)}}}-{f{{\left({u}\right)}}}-{f{{\left({v}\right)}}},{f{{\left({u}+{v}\right)}}}-{f{{\left({u}\right)}}}-{f{{\left({v}\right)}}}⟩}$$\( \mathrm{=⟨{f{{\left({u}+{v}\right)}}},{f{{\left({u}+{v}\right)}}}⟩-{2}⟨{f{{\left({u}\right)}}},{f{{\left({u}+{v}\right)}}}⟩+{2}⟨{f{{\left({u}\right)}}},{f{{\left({v}\right)}}}⟩-{2}⟨{f{{\left({v}\right)}}},{f{{\left({u}+{v}\right)}}}⟩+⟨{f{{\left({u}\right)}}},{f{{\left({u}\right)}}}⟩+⟨{f{{\left({v}\right)}}},{f{{\left({v}\right)}}}⟩-{2}⟨{f{{\left({u}\right ... See the full answer