Q8.Consider 3, w E R? as below, where w is thereflection of j across the -axis. (Note: These vectors have nonzero x-and Y-components, i.e., they are not on the axes.)yW(a) If Av = ww and Aw = i, show that A is reflection across the x-1axis, i.e., A =-1(b) If Av = w and Aw =-i, show that A is similar to a rotationmatrix.

Given that v→,w⇀∈ℝ2 be two vectors in such that w⇀ is the reflection of v⇀ across x axis. Let us…

Answered: Q8. Consider i, w E R² as below, where…

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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Q8.
Consider 3, w E R? as below, where w is the
reflection of j across the -axis. (Note: These vectors have nonzero x-
and Y-components, i.e., they are not on the axes.)
y
W
(a) If Av = w
w and Aw = i, show that A is reflection across the x-
1
axis, i.e., A =
-1
(b) If Av = w and Aw =
-i, show that A is similar to a rotation
matrix.

Expert Solution

part a

Given that $\vec{v}, \overset{⇀}{w} \in ℝ^{2}$ be two vectors in such that $\overset{⇀}{w}$ is the reflection of $\overset{⇀}{v}$ across $x$ axis.

Let us consider $\overset{⇀}{v} = (x, y), \overset{⇀}{w} = (x, - y)$

Now, suppose $A = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$ , where $a$ and $b$ are constants to be determined. It is given $A \overset{⇀}{v} = \overset{⇀}{w}$ and $A \overset{⇀}{w} = \overset{⇀}{v}$

$\begin{array}{rcl} A \overset{⇀}{v} & = & \overset{⇀}{w} \\ \Rightarrow & [\begin{matrix} a & 0 \\ 0 & b \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} x \\ - y \end{matrix}] \\ \Rightarrow & [\begin{matrix} a x \\ b y \end{matrix}] = [\begin{matrix} x \\ - y \end{matrix}] \\ \Rightarrow & a = 1, b = - 1 \end{array}$

Similarly, from $A \overset{⇀}{w} = \overset{⇀}{v}$ , we have $a = 1, b = - 1$

Hence, $A$ is given by

$A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$

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