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 10E from Chapter 1.9 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 10E
Chapter:
Problem:
In Exercises 1–10, assume that T is a linear transformation. Find the standard matrix of T. T : ℝ2 → ℝ2 first reflects points through the horizontal x1-axis and then rotates points –π/2 radians
Step-by-Step Solution
Given Information
We are given with a linear transformation $T:{R^2} \to {R^2}$ that reflects through the vertical $x2_$ axis and then rotates points $\dfrac{\pi }{2}$ radians.
We have to find the standard matrix of transformation. The standard vectors in M are: \[{e_1} = \left[ {\begin{array}{*{20}{l}} 1\\ 0 \end{array}} \right],{e_2} = \left[ {\begin{array}{*{20}{l}} 0\\ 1 \end{array}} \right]\]
Step-1: The transformation of $e_1$
\[\begin{array}{l} {T_1}\left( {{e_1}} \right) = {T_1}\left( {\left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}} { - 1}\\ 0 \end{array}} \right]\\ {T_2}\left( {{T_1}\left( {{e_1}} \right)} \right) = {T_1}\left( {\left[ {\begin{array}{*{20}{c}} { - 1}\\ 0 \end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}} 0\\ { - 1} \end{array}} \right] \end{array}\]
Step-2: The transformation of $e_2$
\[\begin{array}{l} {T_1}\left( {{e_2}} \right) = {T_1}\left( {\left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]\\ {T_2}\left( {{T_1}\left( {{e_2}} \right)} \right) = {T_1}\left( {\left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}} { - 1}\\ 0 \end{array}} \right] \end{array}\]
Step-3: The standard matrix
The standard matrix is: \[ \begin{aligned} A & = \left[ T \left( e _ { 1 } \right) , T \left( e _ { 2 } \right) \right] \\ & = \left[ \begin{array} { c c } { 0 } & { - 1 } \\ { - 1 } & { 0 } \end{array} \right] \end{aligned} \]
\[A = \left[ {\begin{array}{*{20}{c}} 0&{ - 1}\\ { - 1}&0 \end{array}} \right]\]
We are given with a linear transformation $T:{R^2} \to {R^2}$ that reflects through the vertical $x2_$ axis and then rotates points $\dfrac{\pi }{2}$ radians.
We have to find the standard matrix of transformation. The standard vectors in M are: \[{e_1} = \left[ {\begin{array}{*{20}{l}} 1\\ 0 \end{array}} \right],{e_2} = \left[ {\begin{array}{*{20}{l}} 0\\ 1 \end{array}} \right]\]
Step-1: The transformation of $e_1$
\[\begin{array}{l} {T_1}\left( {{e_1}} \right) = {T_1}\left( {\left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}} { - 1}\\ 0 \end{array}} \right]\\ {T_2}\left( {{T_1}\left( {{e_1}} \right)} \right) = {T_1}\left( {\left[ {\begin{array}{*{20}{c}} { - 1}\\ 0 \end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}} 0\\ { - 1} \end{array}} \right] \end{array}\]
Step-2: The transformation of $e_2$
\[\begin{array}{l} {T_1}\left( {{e_2}} \right) = {T_1}\left( {\left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]\\ {T_2}\left( {{T_1}\left( {{e_2}} \right)} \right) = {T_1}\left( {\left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}} { - 1}\\ 0 \end{array}} \right] \end{array}\]
Step-3: The standard matrix
The standard matrix is: \[ \begin{aligned} A & = \left[ T \left( e _ { 1 } \right) , T \left( e _ { 2 } \right) \right] \\ & = \left[ \begin{array} { c c } { 0 } & { - 1 } \\ { - 1 } & { 0 } \end{array} \right] \end{aligned} \]
\[A = \left[ {\begin{array}{*{20}{c}} 0&{ - 1}\\ { - 1}&0 \end{array}} \right]\]