Q33E Give an example of a linear tran... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q33E (page 120)

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

Short Answer

Expert verified

The required linear transformation is,

$A \overset{鈫�}{v} = (x + 2 y + 3 z, 鈥� 鈥� 0, 鈥� 鈥� 0)$

Step by step solution

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

The kernel of linear transformationis defined as follows:

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

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

To give an example of a linear transformation

We require linear transformation $A$ such that

$A \overset{鈫�}{v} = 0$ ,where the plane is $\overset{鈫�}{v} = (x, 鈥� y, 鈥� z) : 鈥� x + 2 y + 3 z = 0$ is ${鈩�}^{3}$ 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,

$A \overset{鈫�}{v} = [\begin{matrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] 路 [\begin{matrix} x \\ y \\ z \end{matrix}] 鈬� A \overset{鈫�}{v} = (x + 2 y + 3 z, 鈥� 鈥� 0, 鈥� 鈥� 0)$

where, $\overset{鈫�}{v} = (x, y, z) : 鈥� x + 2 y + 3 z = 0$

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91影视

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

Short Answer

Step by step solution

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

To give an example of a linear transformation

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91影视

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

Short Answer

Step by step solution

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

To give an example of a linear transformation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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