Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Give an example of a linear transformation whose kernel is the plane $x + 2 y + 3 z = 0$ in $ℝ^{3}$ .

The required linear transformation is,

$A \vec{v} = (x + 2 y + 3 z, 0, 0)$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step by Step Solution: Step 1: To define kernel of linear transformation

The kernel of linear transformation is defined as follows:

The kernel of a linear transformation $T (\vec{x}) = A \vec{x}$ from $ℝ^{m}$ to $ℝ^{n}$ consists of all zeros of the transformation, i.e., the solutions of the equations $T (\vec{x}) = A \vec{x} = 0$

It is denoted by ker $(T)$ or ker $(A)$

Step 2: To give an example of a linear transformation

We require linear transformation $A$ such that

$A \vec{v} = 0$ ,where the plane is $\vec{v} = (x, y, z) : x + 2 y + 3 z = 0$ is $ℝ^{3}$ is the kernel of the transformation.

Since we know that the dot product of the required vector with normal vector of given plane given by,

$[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ ,is equal to 0, the required linear transformation is,

$A \vec{v} = [\begin{matrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] \cdot [\begin{matrix} x \\ y \\ z \end{matrix}] \Rightarrow A \vec{v} = (x + 2 y + 3 z, 0, 0)$

where, $\vec{v} = (x, y, z) : x + 2 y + 3 z = 0$

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Linear Algebra With Applications

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Short Answer

Give an example of a linear transformation whose kernel is the plane $x + 2 y + 3 z = 0$ in $ℝ^{3}$ .

Step by Step Solution

Step by Step Solution: Step 1: To define kernel of linear transformation

Step 2: To give an example of a linear transformation

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Linear Algebra With Applications

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Short Answer

Give an example of a linear transformation whose kernel is the plane x+2y+3z=0in ℝ3.

Step by Step Solution

Step by Step Solution: Step 1: To define kernel of linear transformation

Step 2: To give an example of a linear transformation

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Give an example of a linear transformation whose kernel is the plane $x + 2 y + 3 z = 0$ in $ℝ^{3}$ .