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MTH 4100: Final Practice 1

Problem 1

Consider the following four vectors in \( \mathbb R^4 \): \[ \overrightarrow m=\left[\begin{array}{c} 2\newline 0 \newline 0 \newline 2\end{array}\right], \quad \overrightarrow n= \left[\begin{array}{c} 0\newline 0 \newline 2 \newline 0\end{array}\right], \quad \overrightarrow p=\left[\begin{array}{c} 0\newline 1 \newline 0 \newline -2\end{array}\right], \quad \mbox{and}\quad \overrightarrow q=\left[\begin{array}{c} 6\newline -1 \newline 4 \newline 8\end{array}\right].\]

  • (a) Determine the real numbers \( \alpha \), \( \beta \), and \( \gamma \) such that \( \overrightarrow q=\alpha \overrightarrow m+\beta\overrightarrow n+\gamma\overrightarrow p \).

  • (b) Does the vector \( \overrightarrow q \) belong to the vector space \( \displaystyle V=\mbox{span}\left\{\overrightarrow m,\overrightarrow n,\overrightarrow p\right\} \)?

Problem 2

Does there exist a linear transformation \( L:\mathbb R^3\to\mathbb R^2 \) such that \[ L\left[\begin{array}{c} 3\newline 7\newline0\end{array}\right]=\left[\begin{array}{c} 3\newline 1\end{array}\right], \quad L\left[\begin{array}{c} 4\newline 3\newline5\end{array}\right]=\left[\begin{array}{c} 5\newline 6\end{array}\right], \quad\mbox{and}\quad L\left[\begin{array}{c} -1\newline 4\newline -5\end{array}\right]=\left[\begin{array}{c} -2\newline 0\end{array}\right]?\]

Problem 3

Find the real number \( x \) for which the matrix \( \left[\begin{array}{ccc} x & -2&0\newline 2 & 5&0\newline 0&0&\frac1{19}\end{array}\right] \) is singular.

Problem 4

Assume that \( L:\mathcal P_2\to\mathcal P_2 \) is the linear operator whose matrix with respect to the basis \( B=\left\{1,X,X^2\right\} \) is \[ \mbox{Rep}_{B, B}(L)=\left[\begin{array}{ccc} 2 & 3& 12\newline 5 & 4& -5\newline 5 & 3& 10\end{array}\right].\] Determine whether \( 1+X^2 \) is an eigenvector of \( L \).

Problem 5

  • (a) Find the eigenvalues and the eigenvectors of the matrix \( A=\left[\begin{array}{cc} 2 & 1\newline -6 & 7\end{array}\right] \).

  • (b) Determine \( A^n \), where \( A \) is the matrix from part (a).

  • (c) Determine the formulas for general terms of the sequences \( (a_n) \) and \( (b_n) \) that are defined recursively as \( a_1=2 \), \( b_1=9 \), and for \( n\geq 1 \): \begin{eqnarray*} a_{n+1}&=& 2a_n+ b_n\newline b_{n+1}&=& -6a_n+7b_n. \end{eqnarray*}

Problem 6

Assume that \( \Phi: V\times V\to\mathbb R \) is a bilinear form. Assume that \( \overrightarrow u \) and \( \overrightarrow v \) are two vectors from \( V \) and that \[ \Phi\left(\overrightarrow u, \overrightarrow v\right)=7,\quad \Phi\left(\overrightarrow v, \overrightarrow u\right)=8, \quad \Phi\left(\overrightarrow u, 2\overrightarrow u\right)=2, \quad\mbox{and}\quad \Phi\left(5\overrightarrow v, \overrightarrow v\right)=5.\] Determine \( \Phi\left(2\overrightarrow u+2\overrightarrow v, \overrightarrow u-5\overrightarrow v\right) \).

Problem 7

The vectors \( \overrightarrow u \) and \( \overrightarrow v \) are eigenvectors of a linear operator \( L:V\to V \) such that \( \overrightarrow u \) corresponds to the eigenvalue \( 2 \) and \( \overrightarrow v \) corresponds to the eigenvalue \( 7 \). Find all vectors from \( \mbox{span }\left\{\overrightarrow u, \overrightarrow v\right\} \) that are also eigenvectors of \( L \).

Problem 8

Assume that \( \Phi: \mathbb R^{k}\times \mathbb R^{k}\to \mathbb R \) is a bilinear form, and assume that \( L \) is a matrix of format \( k\times k \) that for every two vectors \( \overrightarrow u, \overrightarrow v\in \mathbb R^{k} \) satisfies \[ \Phi\left( L\overrightarrow u, \overrightarrow v\right)= \Phi\left(\overrightarrow u, L\overrightarrow v\right).\] If \( F \) is the matrix corresponding to the bilinear form \( \Phi \), prove that \( FL=L^TF \).