Solve the Problem $\mathrm { I } _ { 3 } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$

Multiple Choice

Solve the problem.
-The columns of $\mathrm { I } _ { 3 } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$ are $\mathbf { e } _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right] , \mathbf { e } _ { 2 } = \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right] , \mathbf { e } _ { 3 } = \left[ \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right]$ .
Suppose that $\mathrm { T }$ is a linear transformation from $R ^ { 3 }$ into $\mathcal { R } ^ { 2 }$ such that
$\mathrm { T } \left( \mathbf { e } _ { 1 } \right) = \left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] , \mathrm { T } \left( \mathbf { e } _ { 2 } \right) = \left[ \begin{array} { l } 5 \\ 0 \end{array} \right]$ , and $\mathrm { T } \left( \mathbf { e } _ { 3 } \right) = \left[ \begin{array} { r } - 5 \\ 1 \end{array} \right]$
Find a formula for the image of an arbitrary $x = \left[ \begin{array} { l } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{array} \right]$ in $R ^ { 3 }$ .

A)
$T \left[ \begin{array} { l } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{array} \right] = \left[ \begin{array} { l } 3 x _ { 1 } - 2 x _ { 2 } \\ 5 x _ { 1 } \end{array} \right]$
B)
$T \left[ \begin{array} { l } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{array} \right] = \left[ \begin{array} { l } 3 x _ { 1 } - 2 x _ { 2 } \\ 5 x _ { 1 } \\ 5 x _ { 2 } + x _ { 3 } \end{array} \right]$
C)
$$T \left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 }\end{array} \right] = \left[ \begin{array} { l } 3 x _ { 1 } + 5 x _ { 2 } - 5 x _ { 3 } \\5 x _ { 1 } \\- 2 x _ { 1 } + x _ { 3 }\end{array} \right]$$
D)
$$T \left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 }\end{array} \right] = \left[ \begin{array} { l l } 3 x _ { 1 } + 5 x _ { 2 } - 5 x _ { 3 } \\- 2 x _ { 1 } + x _ { 3 }\end{array} \right]$$