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Home Question Bank Other Archive 2024 December 07

Other Questions from Dec 07,2024

Browse the Other Q&A Archive for Dec 07,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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A matrix \( A \) has the following eigenpairs. Complete the eigenpairs. \( \left(\lambda=\square,\left[\begin{array}{c}3-6 i \\ 1\end{array}\right]\right) \quad\left(\lambda=4-2 i,\left[\begin{array}{l}\square \\ \square\end{array}\right]\right) \) Find the remaining eigenvalue for matrix \( A=\left[\begin{array}{rr}3 & -5 \\ 8 & -9\end{array}\right] \) \( \lambda=-3+2 i \). The matrix \( A=\left[\begin{array}{cc}4 & -2 \\ 1 & 1\end{array}\right] \) has eigenvalue \( \lambda_{1}=3 \) with corresponding eigenvector \( \left[\begin{array}{l}2 \\ 1\end{array}\right] \) and eigenvalue \( \lambda_{2}=2 \) with corresponding eigenvector \( \left[\begin{array}{l}1 \\ 1\end{array}\right] \). Use this information to fill in the following matrices for the diagonalization of \( A \). \( A=P D P^{-1}=\left[\begin{array}{ll}\text { Ex: } 5 & \text { a } \\ \square & \end{array}\right]\left[\begin{array}{cc}\square & 0 \\ 0 & 2\end{array}\right]\left[\begin{array}{cc}1 & -1 \\ -1 & 2\end{array}\right] \) Find the eigenvalue(s) of the linear transformation \( \mathcal{P}_{2} \rightarrow \mathcal{P}_{2} \) given by \( T(p(x))=p(-x) \) \[ \begin{array}{l}\square \lambda=1 \\ \square \lambda=-1\end{array} \] Use the equation \( A \mathbf{x}=\lambda \mathbf{x} \) to determine which of the following is an eigenpair for the linear transformation \( T: \mathcal{P}_{1} \rightarrow \mathcal{P}_{1} \) given by \( T(a x+b)=(7 a-6 b) x+(4 a-3 b) \) \( \square(\lambda=2,2 x+1) \) \( \square(\lambda=1, x+1) \) \( \square(\lambda=3,3 x+2) \) Find the discrete Fourier transform of the sequence \( x=\langle x(0), x(1), x(2), x(3))= \) \( (1,-1,1,-1) \). Find the Fourier series for the function \( f(x)=|x| \) defined in \( -\pi<x<\pi \). 4. Determine los valores propios \( y \) vectores propios de la matriz \( \mathbb{A} \) dada por \[ \mathbb{A}=\left[\begin{array}{ccc}1 & 2 & -1 \\ 1 & 0 & 1 \\ 4 & -4 & 5\end{array}\right] \] Además, para cada valor propio \( \lambda \) de \( \mathbb{A} \) verifique que la siguiente igualdad se cumple \[ \mathbb{A} x=\lambda x \] Para ello, tome en particular a un vector propio \( x \) de \( \mathbb{A} \) asociado al valor propio \( \lambda \). 5- (a) [Byron-Fuller 3.16] Show that the eigenvalues of \[ M=\left[\begin{array}{ccc}3 & 5 & 8 \\ -6 & -10 & -16 \\ 4 & 7 & 11\end{array}\right] \] are \( \lambda=0,1,3 \), and that the corresponding eigenvectors are \[ \left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right], \quad\left[\begin{array}{c}1 \\ -2 \\ 1\end{array}\right], \quad \text { and } \quad\left[\begin{array}{c}4 \\ -8 \\ 5\end{array}\right] . \] Construct a diagonalizing matrix \( P \), prove that its inverse exists, compute its inverse, and verify that \( P^{-1} M P \) is diagonal with the eigenvalues on the diagonal. Note that det \( M=0 \). Is it true that any diagonalizable matrix with an eigenvalue equal to zero is singular? (b) What are the kernel and image of \( M \) ? 5- (a) [Byron-Fuller 3.16] Show that the eigenvalues of \[ M=\left[\begin{array}{ccc}3 & 5 & 8 \\ -6 & -10 & -16 \\ 4 & 7 & 11\end{array}\right] \] are \( \lambda=0,1,3 \), and that the corresponding eigenvectors are \[ \left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right], \quad\left[\begin{array}{c}1 \\ -2 \\ 1\end{array}\right], \quad \text { and }\left[\begin{array}{c}4 \\ -8 \\ 5\end{array}\right] \] Construct a diagonalizing matrix \( P \), prove that its inverse exists, compute its inverse, and verify that \( P^{-1} M P \) is diagonal with the eigenvalues on the diagonal. Note that det \( M=0 \). Is it true that any diagonalizable matrix with an eigenvalue equal to zero is singular? (b) What are the kernel and image of \( M \) ? 2. Sean \( S=\left\{v_{1}, v_{2}, v_{3}\right\} \) y \( T=\left\{w_{1}, w_{2}, w_{3}\right\} \) bases para \( \mathbb{R}^{3} \) donde \[ v_{1}=(1,0,1), \quad v_{2}=(-1,0,0), \quad v_{3}=(0,1,2) \] \[ w_{1}=(-1,1,0), \quad w_{2}=(1,2,-1), \quad w_{3}=(0,1,0) \] y Además, sea \( v \in \mathbb{R}^{3} \) con \( v=(1,3,8) \). i) Determine el vector de coordenadas de \( v \) con respecto a las bases \( S \) y \( T \) ii) Calule la matriz transición \( P_{S \leftarrow T} \) de la base T a la base \( S \) iii) Verifique que \( [v] S=P_{S \leftarrow T}[v]_{T} \)
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