Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Consider a nonzero vector $\vec{υ}$ in $ℝ^{3}$ . Using a geometric argument, describe the kernel of the linear transformation $T$ from $ℝ^{3}$ to $ℝ^{3}$ given by,
$T (\vec{x}) = \vec{υ} \times \vec{x}$
See Definition A.9 in the Appendix.

The kernel is a line in space spanned by vector $\vec{υ}$

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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 $(A)$

The definition A.9 cross product in $ℝ^{3}$ is given as follows:

The cross product $\vec{υ} \times \vec{ω}$ of two vectors role="math" localid="1659527572946" $\vec{υ}$ and $\vec{ω}$ in $ℝ^{3}$ is the vector in $ℝ^{3}$ with three properties as follows:

$\vec{υ} \times \vec{ω}$ is orthogonal to both $\vec{υ}$ and $\vec{ω}$ .
$||\vec{υ} \times \vec{ω}|| = ||\vec{υ}|| \sin θ ||\vec{ω}||$ , where $θ$ is angle between $\vec{υ}$ and $\vec{ω}$ with $0 ⩽ θ ⩽ π$ .
The direction of $\vec{υ} \times \vec{ω}$ follows the right-hand rule.

We have given the linear transformation $T : ℝ^{3} \to ℝ^{3}$ defined by $T (\vec{x}) = \vec{υ} \times \vec{x}$ ,

Step 2: To describe the kernel of the linear transformation

With the reference of definition in step 1, we have,

$T (\vec{x}) = \vec{υ} \times \vec{x} = 0^U \vec{x} = λ \vec{υ}, λ^I R$ is any scalar, which means that the cross product is zero.

This implies that kernel is a line in space spanned by vector $\vec{υ}$

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Linear Algebra With Applications

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Short Answer

Consider a nonzero vector $\vec{υ}$ in $ℝ^{3}$ . Using a geometric argument, describe the kernel of the linear transformation $T$ from $ℝ^{3}$ to $ℝ^{3}$ given by,
$T (\vec{x}) = \vec{υ} \times \vec{x}$
See Definition A.9 in the Appendix.

Step by Step Solution

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

Step 2: To describe the kernel of the linear transformation

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Linear Algebra With Applications

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Short Answer

Consider a nonzero vector υ→in ℝ3 . Using a geometric argument, describe the kernel of the linear transformation Tfrom ℝ3to ℝ3 given by,T(x→)=υ→×x→ See Definition A.9 in the Appendix.

Step by Step Solution

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

Step 2: To describe the kernel of the linear transformation

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Pure Maths

Consider a nonzero vector $\vec{υ}$ in $ℝ^{3}$ . Using a geometric argument, describe the kernel of the linear transformation $T$ from $ℝ^{3}$ to $ℝ^{3}$ given by,
$T (\vec{x}) = \vec{υ} \times \vec{x}$
See Definition A.9 in the Appendix.