Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 9E from Chapter 1.9 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 9E
Chapter:
Problem:
In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. T : ℝ2 → ℝ2 first performs a horizontal shear that transforms e2 into e2 + 2e1 (leaving e1 unchanged) and then reflects points through the line x2 = –x1.
Step-by-Step Solution
Given Information
We are given that the transformation $T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$. We have to find the standard matrix of the transformation
The standard vectors are: \[{e_1} = \left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right];\,\,{e_2} = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right];\]
Step-1: Horizontal Shear
\[\begin{array}{l} T\left( {{e_1}} \right) = {e_1} = \left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right]\\ T\left( {{e_2}} \right) = {e_2} + 2{e_1} = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right] - 2\left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 2}\\ 1 \end{array}} \right] \end{array}\]
Step-2: Reflection through ${x_2} = - {x_1}$
\[\begin{array}{l} {T_2}\left( {T\left( {{e_1}} \right)} \right) = {T_2}\left( {{e_1}} \right) = \left[ {\begin{array}{*{20}{c}} 0\\ { - 1} \end{array}} \right]\\ {T_2}\left( {T\left( {{e_2}} \right)} \right) = \left[ {\begin{array}{*{20}{c}} { - 1}\\ 2 \end{array}} \right] \end{array}\] Therefore the standard matrix for given transformation is:
\[ A = \left[ T \left( \mathbf { e } _ { 1 } \right) T \left( \mathbf { e } _ { 2 } \right) \right] _ { 2 \times 2 } = \left[ \begin{array} { c c } { 0 } & { - 1 } \\ { - 1 } & { 2 } \end{array} \right] \]
We are given that the transformation $T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$. We have to find the standard matrix of the transformation
The standard vectors are: \[{e_1} = \left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right];\,\,{e_2} = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right];\]
Step-1: Horizontal Shear
\[\begin{array}{l} T\left( {{e_1}} \right) = {e_1} = \left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right]\\ T\left( {{e_2}} \right) = {e_2} + 2{e_1} = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right] - 2\left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 2}\\ 1 \end{array}} \right] \end{array}\]
Step-2: Reflection through ${x_2} = - {x_1}$
\[\begin{array}{l} {T_2}\left( {T\left( {{e_1}} \right)} \right) = {T_2}\left( {{e_1}} \right) = \left[ {\begin{array}{*{20}{c}} 0\\ { - 1} \end{array}} \right]\\ {T_2}\left( {T\left( {{e_2}} \right)} \right) = \left[ {\begin{array}{*{20}{c}} { - 1}\\ 2 \end{array}} \right] \end{array}\] Therefore the standard matrix for given transformation is:
\[ A = \left[ T \left( \mathbf { e } _ { 1 } \right) T \left( \mathbf { e } _ { 2 } \right) \right] _ { 2 \times 2 } = \left[ \begin{array} { c c } { 0 } & { - 1 } \\ { - 1 } & { 2 } \end{array} \right] \]