Answer
a) The matrix corresponding to the linear transformation $T$ is given by:
$\left[ {\begin{array}{*{20}{c}} { 1}&{0}&0\\ 0&{1}&0\\ 0&0&{-1} \end{array}} \right]$
b) The image of $v=\left[ {\begin{array}{*{20}{c}} 3\\ {2}\\2 \end{array}} \right]$ is $\left[ {\begin{array}{*{20}{c}} { 3}\\ { 2}\\-2 \end{array}} \right] $
c) See image.
Work Step by Step
a) Take the standard basis for $R^3,$ which is $\{(1,0,0),(0,1,0),(0,0,1)\}=\{e_1,e_2,e_3\}$.
We are given
$T(x,y,z)=(x,y,-z)$
Then, we have
$T(1,0,0)=(1,0,0)\\ T(0,1,0)=(0,1,0)\\ T(0,0,1)=(0,0,-1)$
Therefore, the matrix corresponding to the linear transformation $T$ is given by:
$A=[T(e_1)$ $T(e_2)$ $T(e_3)]$ $=\left[ {\begin{array}{*{20}{c}} { 1}&{0}&0\\ 0&{1}&0\\ 0&0&{-1} \end{array}} \right]$
b) Hence, the image of $v=\left[ {\begin{array}{*{20}{c}} 3\\ {2}\\2 \end{array}} \right]$ is
$\begin{array}{l} T(v) = Av\\ \Rightarrow T\left[ {\begin{array}{*{20}{c}} 3\\ {2}\\2 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { 1}&{0}&0\\ 0&{1}&0\\ 0&0&{-1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 3\\ {2}\\2 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { 3}\\ { 2}\\-2 \end{array}} \right] \end{array}$
c) See image.
