Question

Give an example of a linear transformation whose kernel is the line spanned by $\left[\begin{array}{c}-1\\ 1\\ 2\end{array}\right]$in${\mathbf{ℝ}}^{\mathbf{3}}$

Short answer

The required linear transformation is,$A=\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& -1\\ 0& 0& 0\end{array}\right]$.

Step-by-step solution

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

Thekernel 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 $\left(A\right)$.

The given vector is,$\left[\begin{array}{c}-1\\ 1\\ 2\end{array}\right]$in${\mathrm{ℝ}}^3$,which is the plane$-x+y+2z=0$

To give an example of a linear transformation

Consider the vectors ${\overrightarrow{v}}_1=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]$and ${\overrightarrow{v}}_2=\left[\begin{array}{c}-1\\ 1\\ -1\end{array}\right]$

These vectors are non-collinear.

Therefore, it can be represented in the matrix form as:

$\left[\begin{array}{ccc}1& 1& 0\\ -1& 1& -1\\ 0& 0& 0\end{array}\right]$

where, the rows of matrix are the above vectors, which is the required linear transformation.