Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
We have solutions for your book!
See our solution for Question 3E from Chapter 1.9 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 3E
Chapter:
Problem:
In Exercises, assume that T is a linear transformation. Find the standard matrix of T...
Step-by-Step Solution
Step 1
Let the Vectors are:\[{{\bf{e}}_{\bf{1}}} = \left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right];\,\,\,{{\bf{e}}_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right]\]Given Transformation (see Figure 1 in the section 9.1 of textbook):\[T:{{\rm{R}}^2} \to {{\rm{R}}^2}::T\left( {{{\bf{e}}_1}} \right) = \left[ {\begin{array}{*{20}{c}}{\cos \phi }\\{\sin \phi }\end{array}} \right],\,\,:T\left( {{{\bf{e}}_2}} \right) = \left[ {\begin{array}{*{20}{c}}{ - \sin \phi }\\{\cos \phi }\end{array}} \right]\,\]We have to find the transformation matrix for the above transformation.
Step 2: The Rotation Angle
\[\begin{array}{l}\phi = \dfrac{{3\pi }}{2}\\\\\phi = \dfrac{{3\pi }}{2}{\kern 1pt} \dfrac{{180^\circ }}{\pi }\\\\\phi = 270^\circ \end{array}\]
Step 3: The Transformation Matrix
\[\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}{\cos \phi }&{ - \sin \phi }\\{\sin \phi }&{\cos \phi }\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{\cos 270^\circ }&{ - \sin 270^\circ }\\{\sin 270^\circ }&{\cos 270^\circ }\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}0&1\\{ - 1}&0\end{array}} \right]\end{array}\]
ANSWER
\[A = \left[ {\begin{array}{*{20}{c}}0&1\\{ - 1}&0\end{array}} \right]\]
Let the Vectors are:\[{{\bf{e}}_{\bf{1}}} = \left[ {\begin{array}{*{20}{c}}0\\1\end{array}} \right];\,\,\,{{\bf{e}}_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right]\]Given Transformation (see Figure 1 in the section 9.1 of textbook):\[T:{{\rm{R}}^2} \to {{\rm{R}}^2}::T\left( {{{\bf{e}}_1}} \right) = \left[ {\begin{array}{*{20}{c}}{\cos \phi }\\{\sin \phi }\end{array}} \right],\,\,:T\left( {{{\bf{e}}_2}} \right) = \left[ {\begin{array}{*{20}{c}}{ - \sin \phi }\\{\cos \phi }\end{array}} \right]\,\]We have to find the transformation matrix for the above transformation.
Step 2: The Rotation Angle
\[\begin{array}{l}\phi = \dfrac{{3\pi }}{2}\\\\\phi = \dfrac{{3\pi }}{2}{\kern 1pt} \dfrac{{180^\circ }}{\pi }\\\\\phi = 270^\circ \end{array}\]
Step 3: The Transformation Matrix
\[\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}{\cos \phi }&{ - \sin \phi }\\{\sin \phi }&{\cos \phi }\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}{\cos 270^\circ }&{ - \sin 270^\circ }\\{\sin 270^\circ }&{\cos 270^\circ }\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{c}}0&1\\{ - 1}&0\end{array}} \right]\end{array}\]
ANSWER
\[A = \left[ {\begin{array}{*{20}{c}}0&1\\{ - 1}&0\end{array}} \right]\]