The effect of the linear transformation T: R2 R?with matrix A =6 is to reflect each vectorin the x-axis. By arguing geometrically, determine alleigenvalues and eigenvectors of A.

Given : A=100-1 To Determine : Eigenvalues and eigenvector geometrically .

Answered: The effect of the linear transformation…

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

The effect of the linear transformation T: R2 R?
with matrix A =
6 is to reflect each vector
in the x-axis. By arguing geometrically, determine all
eigenvalues and eigenvectors of A.

Expert Solution

Step 1

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

To Determine : Eigenvalues and eigenvector geometrically .

Step 2

Consider a Point P(x ,y) =(x , y)

$T (x, y) = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} x \\ - y \end{matrix}]$

$\Rightarrow T (x, y) = (x, - y)$

Hence T is reflection of point along x-axis

Now Consider point along x-axis (x , 0)

then $T (x, 0) = (x, 0)$

It is Fixed Points

Hence point in the x axis are considered as 1-eigen space and hence the 1 is an eigen value for T

Now Consider points on the y-axis (0 , y)

then T(0 , y)=(0 , -y)

Hence y-axis is an eigen space with -1 as the eigen value for T

Hence Eigen value is 1 and -1

Now , we find eigen vector corresponding eigen value 1

$(A - 1 I) X = 0$ where $X = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]$

$[\begin{matrix} 0 & 0 \\ 0 & - 2 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}]$

$\Rightarrow x_{2} = 0$

$X_{1} = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} x_{1} \\ 0 \end{matrix}] = x_{1} [\begin{matrix} 1 \\ 0 \end{matrix}]$

if $x_{1} = 1$

$X_{1} = [\begin{matrix} 1 \\ 0 \end{matrix}]$

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The effect of the linear transformation T: R2 R? with matrix A = 6 is to reflect each vector in the x-axis. By arguing geometrically, determine all eigenvalues and eigenvectors of A.