Answered: Determine the kernel and range of each…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 16CM

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Question

Determine the kernel and range of each of the
following linear operators on R3: L (x) = (x3, x2, x1)T

Expert Solution

Step 1

Null Space: The set of all vectors x such that $A x = 0$ is called the null space of the matrix A. This is

called the kernel of the matrix.

It is given that the linear transformation is $L (x) = {(x_{3}, x_{2}, x_{1})}^{T}$ . We can write the given transformation

as $L (x) = (\begin{matrix} x_{3} \\ x_{2} \\ x_{1} \end{matrix})$ .

Step 2

We can write the given linear transformation in the matrix form as below:

$L (x) = (\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix})$

The null space of the linear transformation is given by

$\begin{array}{rcl} Ker (L) & = & \{(x_{1}, x_{2}, x_{3}) L (x) = 0\} \\ = & \{(x_{1}, x_{2}, x_{3}) {(x_{3}, x_{2}, x_{1})}^{T} = 0\} \\ = & \{0\} \\ ∴ Ker (L) & = & \{0\} \end{array}$

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