Question

1. (8 pts.) Suppose T is a linear transformation and -(1)-(C). --- () - 0 Solve the matrix vector equations below. Explain 2. (8 pts.) Below are a matrix A, its inverse, and a vector b. 1983--013... (1) A = A-1 = 1-4 3 2 -5 7 2 0 1) , and b = x) Determine the value of x. Then solve the matrix vector equation Av = b. [-u-5 il 14 s1 3 70 | 71-3-70 12 x 12 1-2 - 27

Answer 1

Solution: 1.

$\small T$  is a linear transformation and

$\small T\begin{pmatrix} 1\\ 2\\ 4\\ \end{pmatrix}=\begin{pmatrix} -1\\ -2\\ 3\\ \end{pmatrix}$ and $\small T\begin{pmatrix} 2\\ -1\\ -2\\ \end{pmatrix}=\begin{pmatrix} 4\\ 0\\ 6\\ \end{pmatrix}$ .

Consider

$\small T\left ( v_{1} \right )=\begin{pmatrix} 2\\ 4\\ -6\\ \end{pmatrix}=-2\begin{pmatrix} -1\\ -2\\ 3\\ \end{pmatrix}=-2T\begin{pmatrix} 1\\ 2\\ 4\\ \end{pmatrix}=T\begin{pmatrix} -2\\ -4\\ -8\\ \end{pmatrix}$

$\small {\color{DarkRed} \therefore v_{1}=\begin{pmatrix} -2\\ -4\\ -8\\ \end{pmatrix}}$

$\small T\left ( v_{2} \right )=\begin{pmatrix} -5\\ -2\\ -3\\ \end{pmatrix}=\begin{pmatrix} -4\\ 0\\ -6\\ \end{pmatrix}+\begin{pmatrix} -1\\ -2\\ 3\\ \end{pmatrix}=-1\begin{pmatrix} 4\\ 0\\ 6\\ \end{pmatrix}+\begin{pmatrix}= -1\\ -2\\ 3\\ \end{pmatrix}$

  $\small =-1T\begin{pmatrix} 2\\ -1\\ -2\\ \end{pmatrix}+T\begin{pmatrix} 1\\ 2\\ 4\\ \end{pmatrix}=T\begin{pmatrix} -2\\ 1\\ 2\\ \end{pmatrix}+T\begin{pmatrix} 1\\ 2\\ 4\\ \end{pmatrix}=T\begin{pmatrix} -2+1\\ 1+2\\ 2+4\\ \end{pmatrix}=T\begin{pmatrix} -1\\ 3\\ 6\\ \end{pmatrix}$

$\small {\color{Green} \therefore v_{2}=\begin{pmatrix} -1\\ 3\\ 6\\ \end{pmatrix}}$