Question

Give an example of a linear transformation whose kernel is the plane $\mathit{x}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{+}\mathbf{3}\mathit{z}\mathbf{=}\mathbf{0}$in${\mathbf{ℝ}}^{\mathbf{3}}$.

Short answer

The required linear transformation is,

$A\overrightarrow{v}=\left(x+2y+3z, 0, 0\right)$

Step-by-step solution

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

The kernel of linear transformationis 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$\left(A\right)$

To give an example of a linear transformation

We require linear transformation $A$such that

$A\overrightarrow{v}=0$,where the plane is $\overrightarrow{v}=\left(x, y, z\right): x+2y+3z=0$is ${\mathrm{ℝ}}^3$is the kernel of the transformation.

Since we know that the dot product of the required vector with normal vector of given plane given by,

$\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$,is equal to 0, the required linear transformation is,

$$\begin{gathered} A\overrightarrow{v}=\left[\begin{array}{ccc}1& 2& 3\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]·\left[\begin{array}{c}x\\ y\\ z\end{array}\right] \\ \Rightarrow A\overrightarrow{v}=\left(x+2y+3z, 0, 0\right) \end{gathered}$$

where,$\overrightarrow{v}=\left(x,y,z\right): x+2y+3z=0$