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Expert-verified Found in: Page 120 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Consider a nonzero vector $\stackrel{\mathbf{\to }}{\mathbf{\upsilon }}$in ${{\mathbf{ℝ}}}^{{\mathbf{3}}}$ . Using a geometric argument, describe the kernel of the linear transformation ${\mathbit{T}}$from ${{\mathbf{ℝ}}}^{{\mathbf{3}}}$to ${{\mathbf{ℝ}}}^{{\mathbf{3}}}$ given by,${\mathbit{T}}\left(\stackrel{\to }{x}\right){\mathbf{=}}\stackrel{\mathbf{\to }}{\mathbf{\upsilon }}{\mathbf{×}}\stackrel{\mathbf{\to }}{\mathbf{x}}$ See Definition A.9 in the Appendix.

The kernel is a line in space spanned by vector $\stackrel{\to }{\upsilon }$

See the step by step solution

## 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 $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ from ${\mathrm{ℝ}}^{m}$ to ${\mathrm{ℝ}}^{n}$ consists of all zeros of the transformation, i.e., the solutions of the equations $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}=0$.

It is denoted by ker$\left(T\right)$ or ker$\left(A\right)$

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

The cross product $\stackrel{\to }{\upsilon }×\stackrel{\to }{\omega }$ of two vectors role="math" localid="1659527572946" $\stackrel{\to }{\upsilon }$and $\stackrel{\to }{\omega }$ in ${\mathrm{ℝ}}^{3}$ is the vector in ${\mathrm{ℝ}}^{3}$ with three properties as follows:

1. $\stackrel{\to }{\upsilon }×\stackrel{\to }{\omega }$is orthogonal to both $\stackrel{\to }{\upsilon }$and$\stackrel{\to }{\omega }$.
2. $||\stackrel{\to }{\upsilon }×\stackrel{\to }{\omega }||=||\stackrel{\to }{\upsilon }||\mathrm{sin}\theta ||\stackrel{\to }{\omega }||$, where $\theta$ is angle between $\stackrel{\to }{\upsilon }$and $\stackrel{\to }{\omega }$ with $0⩽\theta ⩽\pi$.
3. The direction of $\stackrel{\to }{\upsilon }×\stackrel{\to }{\omega }$ follows the right-hand rule.

We have given the linear transformation $T:{\mathrm{ℝ}}^{3}\to {\mathrm{ℝ}}^{3}$ defined by $T\left(\stackrel{\to }{x}\right)=\stackrel{\to }{\upsilon }×\stackrel{\to }{x}$,

## Step 2: To describe the kernel of the linear transformation

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

$T\left(\stackrel{\to }{x}\right)=\stackrel{\to }{\upsilon }×\stackrel{\to }{x}=0^U\stackrel{\to }{x}=\lambda \stackrel{\to }{\upsilon }, \lambda ^IR$ is any scalar, which means that the cross product is zero.

This implies that kernel is a line in space spanned by vector $\stackrel{\to }{\upsilon }$ ### Want to see more solutions like these? 