text: # Tagged: trigonometry function
  
  ## Problem 567
  
  Prove the following identity for any positive integer $n$.
  $\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}^n=\begin{bmatrix} \cos n\theta & -\sin n\theta\\ \sin n\theta& \cos n\theta \end{bmatrix}.$
  
  ## Problem 550
  
  Consider the $2\times 2$ matrix
  $A=\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix},$ where $\theta$ is a real number $0\leq \theta < 2\pi$.
  
  (a) Find the characteristic polynomial of the matrix $A$.
  
  (b) Find the eigenvalues of the matrix $A$.
  
  (c) Determine the eigenvectors corresponding to each of the eigenvalues of $A$.
  
  ## Problem 533
  
  Consider the complex matrix
  $A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},$ where $x$ is a real number between $0$ and $2\pi$.
  
  Determine for which values of $x$ the matrix $A$ is diagonalizable.
  When $A$ is diagonalizable, find a diagonal matrix $D$ so that $P^{-1}AP=D$ for some nonsingular matrix $P$.
  
  ## Problem 435
  
  Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
  Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
  $\left(\, f\left(\, \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.$ We put
  $V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.$
  
  (a) Prove that the map $f$ is a linear transformation.
  
  (b) Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.
  
  (c) Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.
  (This yields an isomorphism of $\R^2$ and $V$.)
  
  (d) Define a map $g:V \to V$ by
  $g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.$ Prove that the map $g$ is a linear transformation.
  
  (e) Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.
  
  (Kyoto University, Linear Algebra exam problem)
explanation: The provided text contains multiple math problems related to trigonometric functions and linear algebra. Each problem is a distinct question with no provided answers.
  
  - **Problem 567**: Prove a matrix identity involving trigonometric functions.
  - **Problem 550**: Find the characteristic polynomial, eigenvalues, and eigenvectors of a given matrix.
  - **Problem 533**: Determine the values of \( x \) for which a complex matrix is diagonalizable and find the corresponding diagonal matrix.
  - **Problem 435**: Prove properties of a linear transformation and find the matrix representation of a derivative operator.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Tagged: trigonometry function
  
  ## Problem 567
  
  Prove the following identity for any positive integer $n$.
  $\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}^n=\begin{bmatrix} \cos n\theta & -\sin n\theta\\ \sin n\theta& \cos n\theta \end{bmatrix}.$
  
  ## Problem 550
  
  Consider the $2\times 2$ matrix
  $A=\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix},$ where $\theta$ is a real number $0\leq \theta < 2\pi$.
  
  (a) Find the characteristic polynomial of the matrix $A$.
  
  (b) Find the eigenvalues of the matrix $A$.
  
  (c) Determine the eigenvectors corresponding to each of the eigenvalues of $A$.
  
  ## Problem 533
  
  Consider the complex matrix
  $A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},$ where $x$ is a real number between $0$ and $2\pi$.
  
  Determine for which values of $x$ the matrix $A$ is diagonalizable.
  When $A$ is diagonalizable, find a diagonal matrix $D$ so that $P^{-1}AP=D$ for some nonsingular matrix $P$.
  
  ## Problem 435
  
  Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
  Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
  $\left(\, f\left(\, \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.$ We put
  $V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.$
  
  (a) Prove that the map $f$ is a linear transformation.
  
  (b) Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.
  
  (c) Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.
  (This yields an isomorphism of $\R^2$ and $V$.)
  
  (d) Define a map $g:V \to V$ by
  $g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.$ Prove that the map $g$ is a linear transformation.
  
  (e) Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.
  
  (Kyoto University, Linear Algebra exam problem)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text contains multiple math problems related to trigonometric functions and linear algebra. Each problem is a distinct question with no provided answers.
  
  - **Problem 567**: Prove a matrix identity involving trigonometric functions.
  - **Problem 550**: Find the characteristic polynomial, eigenvalues, and eigenvectors of a given matrix.
  - **Problem 533**: Determine the values of \( x \) for which a complex matrix is diagonalizable and find the corresponding diagonal matrix.
  - **Problem 435**: Prove properties of a linear transformation and find the matrix representation of a derivative operator.
  
  Label: multiple-question