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Home Question Bank Other Archive 2024 December 15

Other Questions from Dec 15,2024

Browse the Other Q&A Archive for Dec 15,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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4. Let \( \theta \in \mathbb{R} \) and define a map \( L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \) by \[ L\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right) \] 1 Find all real eigen-values \( \lambda \in \mathbb{R} \) of \( [L]_{\mathcal{E}}^{\mathcal{E}} \), and find all eigen-vectors corresponding to each eigen-values. 5. Let \( \theta \in \mathbb{R} \) and define a map \( L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \) by \[ L\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{ccc}-1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right) \] Find all real eigen-values \( \lambda \in \mathbb{R} \) of \( [L]_{\mathcal{E}}^{\mathcal{E}} \), and find all eigen-vectors corresponding to each eigen-values. 1. If \( u=3 i+6 j-7 k, v=4 i-5 j+6 k \) and \( w=-2 i+3 j-9 k \) then use the algebra of vectors find the followings \( >(3 u-4 v) \bullet(5 w-2 u) \) \( >(2 \mathrm{v}+3 \mathrm{w}) \bullet(4 \mathrm{u}-3 \mathrm{v}) \) \( >(\mathrm{u}+\mathrm{v}+3 \mathrm{w}) \bullet(\mathrm{u}-\mathrm{v}-2 \mathrm{w}) \) \( >(2 \mathrm{u}+\mathrm{v}) \times(\mathrm{v}-2 \mathrm{w}) \) \( >(\mathrm{w}+\mathrm{u}) \times(\mathrm{w}-\mathrm{u}) \) 1. If \( u=3 i+6 j-7 k, v=4 i-5 j+6 k \) and \( w=-2 i+3 j-9 k \) then use the algebra of vectors find the followings \( >(3 u-4 v) \bullet(5 w-2 u) \) \( >(2 \mathrm{v}+3 \mathrm{w}) \bullet(4 \mathrm{u}-3 \mathrm{v}) \) \( >(\mathrm{u}+\mathrm{v}+3 \mathrm{w}) \bullet(\mathrm{u}-\mathrm{v}-2 \mathrm{w}) \) \( >(2 \mathrm{u}+\mathrm{v}) \times(\mathrm{v}-2 \mathrm{w}) \) \( >(\mathrm{w}+\mathrm{u}) \times(\mathrm{w}-\mathrm{u}) \) 4. Let \( \theta \in \mathbb{R} \) and define a map \( L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \) by \[ L\left(\begin{array}{c}x \\ y \\ z\end{array}\right)=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta\end{array}\right)\left(\begin{array}{c}x \\ y \\ z\end{array}\right) \] 1 Find all real eigen-values \( \lambda \in \mathbb{R} \) of \( [L]_{\mathcal{E}}^{\mathcal{E}} \), and find all eigen-vectors corresponding to each eigen-values. 5. Let \( \theta \in \mathbb{R} \) and define a map \( L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \) by \[ L\left(\begin{array}{c}x \\ y \\ z\end{array}\right)=\left(\begin{array}{ccc}-1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right) \] Find all real eigen-values \( \lambda \in \mathbb{R} \) of \( [L]_{\mathcal{E}}^{\mathcal{E}} \), and find all eigen-vectors corresponding to each eigen-values. 4. Let \( \theta \in \mathbb{R} \) and define a map \( L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \) by \[ L\left(\begin{array}{c}x \\ y \\ z\end{array}\right)=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta\end{array}\right)\left(\begin{array}{c}x \\ y \\ z\end{array}\right) \] 1 Find all real eigen-values \( \lambda \in \mathbb{R} \) of \( [L]_{\mathcal{E}}^{\mathcal{E}} \), and find all eigen-vectors corresponding to each eigen-values. 3. Let \( \theta \in \mathbb{R} \) and define a map \( L: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \) by \[ \begin{array}{l}L\binom{x}{y}\end{array}=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)\binom{x}{y} \] Prove that \( \left\{\mathbf{a} \in \mathbb{R}^{2}: L(\mathbf{a})=\mathbf{a}\right\}=\{\mathbf{0}\} \) if \( \theta \notin 2 \pi \mathbb{Z}=\{2 \pi n: n \in \mathbb{Z}\} \) Let a be a nonzero vector in \( \mathbb{R}^{2} \). Define \( \tau_{\mathbf{a}}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \) by \[ \tau_{\mathbf{a}}(\mathbf{x})=-\mathbf{x}+2 \frac{\mathbf{a} \cdot \mathbf{x}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a} \text {. } \] 2. Let a be a nonzero vector of \( \mathbb{R}^{2} \). Answer the following questions. (i) Show that \( \tau_{\mathbf{a}} \) is a linear map. (ii) Write \( \mathbf{a}=\left(a_{1}, a_{2}\right) \) and let \( \mathbf{b}=\left(-a_{2}, a_{1}\right) \in \mathbb{R}^{2} \). Show that \( \mathcal{B}=\{\mathbf{a}, \mathbf{b}\} \) is a basis for \( \mathbb{R}^{2} \). ¿Qué es un numero adimensional?
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