Question

Consider a nonzero vector $\overrightarrow{\mathbf{\upsilon }}$in ${\mathbf{ℝ}}^{\mathbf{3}}$. Using a geometric argument, describe the kernel of the linear transformation $\mathit{T}$from ${\mathbf{ℝ}}^{\mathbf{3}}$to ${\mathbf{ℝ}}^{\mathbf{3}}$given by,

$\mathit{T}\left(\overrightarrow{x}\right)\mathbf{=}\overrightarrow{\mathbf{\upsilon }}\mathbf{\times }\overrightarrow{\mathbf{x}}$

See Definition A.9 in the Appendix.

Short answer

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

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

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

Thedefinition A.9 cross productin ${\mathrm{ℝ}}^3$is given as follows:

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

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

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

To describe the kernel of the linear transformation

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

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

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