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 \( \alpha \in \mathbb{R} \).

UYZJQW The Asker · Other Mathematics

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 \( \alpha \in \mathbb{R} \).

More

Community Answer

D5TVJ3

【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/3Given:  LetU andV be real vector spaces (i.e. vector spaces overR), equipped with scalar products \( \mathrm{⟨⋅,⋅⟩} \) Uand\( \mathrm{⟨⋅,⋅⟩} \)V, 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