Question

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

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

Short answer

Thus, the image of linear transformation is$im\left(T\right)=\overrightarrow{x}:\overrightarrow{x}\cdot \overrightarrow{v}=0$ and the kernel of linear transformation is $kerT=\overrightarrow{x}:\overrightarrow{x}=a\overrightarrow{v},a\in R$.

Step-by-step solution

Determine kernel of the linear transformation.

The cross product of$\overrightarrow{v}$ and$\overrightarrow{x}$ is $\overrightarrow{0}$. When the angle between the two vectors is 0.

Thus, the kernel of the given transformation is the set of$\overrightarrow{v}$ that is parallel to $\overrightarrow{v}$.

Hence, the kernel of linear transformation is $kerT=\left\{\overrightarrow{x}:\overrightarrow{x}=a\overrightarrow{v},a\in !\right\}$.

Determine  image of the linear transformation.

The cross product of$\overrightarrow{v}$ and$\overrightarrow{x}$ is always orthogonal to $\overrightarrow{v}$. Thus, the image of the transformation is the plane orthogonal to or it is written as follows;

$im\left(T\right)=\left\{\overrightarrow{x}:\overrightarrow{x}\cdot \overrightarrow{v}\right\}=0$

Hence, the image of linear transformation is $im\left(T\right)=\left\{\overrightarrow{x}:\overrightarrow{x}\cdot \overrightarrow{v}=0\right\}$.