Answered: 6) Define the linear transformation T:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(ED-E
6) Define the linear transformation T: R³ → R³ by T
a) What is the matrix that represents T relative to the standard basis for R3.
b) Use the result of part (a) and matrix multiplication to find T
X2

Expert Solution

Step 1

$(6)$ Define a linear transformation $T : R^{3} \to R^{3}$ by $T ([\begin{matrix} x \\ y \\ z \end{matrix}]) = [\begin{matrix} x \\ - y \\ z \end{matrix}]$ .

$(a)$ We have to find the matrix that represents $T$ relative to the standard basis for $R^{3}$ .

Take the standard basis for $R^{3}$ is $B = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ .

$\begin{matrix} T ([\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]) = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] = 1 [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] + 0 [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] + 0 [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] \\ T ([\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]) = [\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix}] = 0 [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] + (- 1) [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] + 0 [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] \\ T ([\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] = 0 [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] + 0 [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] + 1 [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] \end{matrix}$