Chapter 6 - Linear Transformations - 6.1 Introduction to Linear Transformations - 6.1 Exercises - Page 301: 41

a) $T\left[ {\begin{array}{*{20}{c}} 1\\ 1\\ 1\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 1}\\ 1\\ 2\\ 1 \end{array}} \right]$ b) Preimage of $\left[ {\begin{array}{*{20}{c}} 1\\ 1\\ 1\\ 1 \end{array}} \right]$ is $\left[ {\begin{array}{*{20}{c}} -1\\ 1\\ \frac{1}{2}\\ 1 \end{array}} \right]$

a) $T\left[ {\begin{array}{*{20}{c}} 1\\ 1\\ 1\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 1}&0&0&0\\ 0&1&0&0\\ 0&0&2&0\\ 0&0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1\\ 1\\ 1\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 1}\\ 1\\ 2\\ 1 \end{array}} \right]$ b) Let the preimage of $\left[ {\begin{array}{*{20}{c}} 1\\ 1\\ 1\\ 1 \end{array}} \right]$ be $\left[ {\begin{array}{*{20}{c}} x\\ y\\ z\\ w \end{array}} \right]$, then $\displaystyle \begin{array}{l} T\left[ {\begin{array}{*{20}{c}} x\\ y\\ z\\ w \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ 1\\ 1\\ 1 \end{array}} \right]\\ \Rightarrow \left[ {\begin{array}{*{20}{c}} { - 1}&0&0&0\\ 0&1&0&0\\ 0&0&2&0\\ 0&0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x\\ y\\ z\\ w \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ 1\\ 1\\ 1 \end{array}} \right]\\ \Rightarrow \left[ {\begin{array}{*{20}{c}} { - x}\\ y\\ {2z}\\ w \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ 1\\ 1\\ 1 \end{array}} \right]\\ \Rightarrow x = - 1,y = 1,z = \frac{1}{2},w = 1 \end{array}$ Hence, the preimage of $\left[ {\begin{array}{*{20}{c}} 1\\ 1\\ 1\\ 1 \end{array}} \right]$ is $\left[ {\begin{array}{*{20}{c}} -1\\ 1\\ \frac{1}{2}\\ 1 \end{array}} \right]$