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MTH 4100: Midterm 2 Practice 1

Problem 1

Determine the inverse of the matrix \( \left[\begin{array}{cccc}11&2&0&0\newline 0&0&3&0\newline 0&0&6&7\newline 27&5&0&0\end{array}\right] \).

Problem 2

Assume that \( L:\mathbb R^3\to\mathbb R^2 \) is a linear transformation such that \[ L\left[\begin{array}{c} -3\newline 6\newline 1\end{array}\right]=\left[\begin{array}{c} 2\newline 1\end{array}\right],\quad L\left[\begin{array}{c} -2\newline 6\newline{1}\end{array}\right]=\left[\begin{array}{c} 1\newline 2\end{array}\right],\quad \mbox{and}\quad L\left[\begin{array}{c} -3\newline 7\newline{1}\end{array}\right]=\left[\begin{array}{c} 3\newline 1\end{array}\right].\] Determine the matrix of \( L \) with respect to the standard bases.

Problem 3

Consider the linear transformation \( L:\mathcal P_2\to\mathcal P_2 \) defined in the following way: \[ L\left(a_0+a_1X+a_2X^2\right)= a_0+a_1\left(X + 5\right)+a_2\left(X + 5\right)^2.\] Represent this linear transformation with respect to the pair of basis \( B \), \( B \), where \( B=\left\{1,X,X^2\right\} \).

Problem 4

Assume that \( U \) is a vector space and \( \left\{\overrightarrow{e_1},\overrightarrow{e_2},\overrightarrow{e_3},\overrightarrow{e_4}\right\} \) its basis. Consider the linear transformation \( L:U\to U \) that satisfies \begin{eqnarray*} L\left(\overrightarrow{e_1}\right)&=&-6 \overrightarrow{e_1}-2\overrightarrow{e_2} -2\overrightarrow{e_4} \newline L\left(\overrightarrow{e_2}\right)&=& \overrightarrow{e_1}+3\overrightarrow{e_2} +\overrightarrow{e_3}\newline L\left(\overrightarrow{e_3}\right)&=& 3\overrightarrow{e_1}+\overrightarrow{e_2} +\overrightarrow{e_4}\newline L\left(\overrightarrow{e_4}\right)&=& 6\overrightarrow{e_1}+18\overrightarrow{e_2} +6\overrightarrow{e_3}\newline \end{eqnarray*} Determine the null space of \( L \).

Problem 5

Let \( V \) be a vector space with basis \( \left\{\overrightarrow{e_1},\overrightarrow{e_2},\overrightarrow{e_3}\right\} \). Assume that \( L:V\to V \) is a linear transformation that satisfies: \begin{eqnarray*} L\left(\overrightarrow{e_1}\right)&=& 3\overrightarrow{e_1}\newline L\left(\overrightarrow{e_2}\right)&=& \overrightarrow{e_1}+3\overrightarrow{e_2}\newline L\left(\overrightarrow{e_3}\right)&=& 3\overrightarrow{e_3}. \end{eqnarray*} Consider the linear transformation \( K:V\to V \) defined in the following way \[ K\left(\overrightarrow v\right)=L^{62}\left(\overrightarrow v\right)=L\left(L\left(L\left(\cdots \left(L\left(\overrightarrow v\right)\right)\cdots\right)\right)\right). \quad \mbox{(The transformation }L\mbox{ is applied }62\mbox{ times.)}\] Determine the matrix of \( K \).

Problem 6

Assume that \( E:U\to V \) and \( F:V\to W \) are two linear transformations such that \( \mathcal R(E)=V \) and \( \mathcal R(F)=W \). Determine \( \mathcal R(F\circ E) \).