Question

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

role="math" localid="1659526111480" $\mathit{T}\mathbf{\left(}\overrightarrow{\mathbf{x}}\mathbf{\right)}\mathbf{=}\overrightarrow{\mathbf{\upsilon }}\mathbf{·}\overrightarrow{\mathbf{x}}$.

Short answer

The kernel of transformation consists of all $\overrightarrow{x}\in {\mathrm{ℝ}}^3$that are orthogonal to vectors$\overrightarrow{\upsilon }$and the image is the whole$\mathrm{ℝ}$.

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$\left(A\right)$.

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

To describe the image of the linear transformation

The image of $T$consists of all vectors represented as:

$\left[{\upsilon }_{1 } {\upsilon }_2 {\upsilon }_3\right]$in${\mathrm{ℝ}}^3$ , which are the column vectors.

Here, ${\upsilon }_i\in \mathrm{ℝ}$which are dependent and any of them spans the whole$\mathrm{ℝ}$ .

To describe the kernel of the linear transformation

The kernel of transformation consists of all $\overrightarrow{x}\in {\mathrm{ℝ}}^3$that are orthogonal to vectors $\overrightarrow{\upsilon }$.

Therefore, their linear combination is 0.