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 4E from Chapter 1.9 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 4E
Chapter:
Problem:
In Exercises, assume that T is a linear transformation. Find the standard matrix of T.T : ℝ2→ ℝ2 rotates points (about the origin) through −π/4 radians (clockwise). [Hint: T(e1) = (l/, −1/).]
Step-by-Step Solution
Given Information
We have to find the standard matrix for a transformation $T:{R^2} \to {R^2}$ that rotates points through $ - \dfrac{\pi }{4}$ radians (clockwise)
Step-1: The rotation of standard matrices
Transformation of standard vector $e_1$:\[T\left( {{e_1}} \right) = T\left( {\left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}}{\cos \phi }\\{ - \sin \phi }\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\cos \left( { - \dfrac{\pi }{4}} \right)}\\{ - \sin \left( { - \dfrac{\pi }{4}} \right)}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }\\{ - 1/\sqrt 2 }\end{array}} \right]\]Transformation of standard vector $e_2$:\[T\left( {{e_2}} \right) = T\left( {\left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}}{\sin \phi }\\{\cos \phi }\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\sin \left( { - \dfrac{\pi }{4}} \right)}\\{\cos \left( { - \dfrac{\pi }{4}} \right)}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }\\{1/\sqrt 2 }\end{array}} \right]\]
Step-2: The standard matrix
The standard matrix is given by combining the transformations of standard vectors
\[A = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }&{1/\sqrt 2 }\\{ - 1/\sqrt 2 }&{1/\sqrt 2 }\end{array}} \right]\]
We have to find the standard matrix for a transformation $T:{R^2} \to {R^2}$ that rotates points through $ - \dfrac{\pi }{4}$ radians (clockwise)
Step-1: The rotation of standard matrices
Transformation of standard vector $e_1$:\[T\left( {{e_1}} \right) = T\left( {\left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}}{\cos \phi }\\{ - \sin \phi }\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\cos \left( { - \dfrac{\pi }{4}} \right)}\\{ - \sin \left( { - \dfrac{\pi }{4}} \right)}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }\\{ - 1/\sqrt 2 }\end{array}} \right]\]Transformation of standard vector $e_2$:\[T\left( {{e_2}} \right) = T\left( {\left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right]} \right) = \left[ {\begin{array}{*{20}{c}}{\sin \phi }\\{\cos \phi }\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\sin \left( { - \dfrac{\pi }{4}} \right)}\\{\cos \left( { - \dfrac{\pi }{4}} \right)}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }\\{1/\sqrt 2 }\end{array}} \right]\]
Step-2: The standard matrix
The standard matrix is given by combining the transformations of standard vectors
\[A = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }&{1/\sqrt 2 }\\{ - 1/\sqrt 2 }&{1/\sqrt 2 }\end{array}} \right]\]