Question

Give an example of a matrixAsuch thatim(A)is the plane with normal vector $\left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]$in${\mathbf{ℝ}}^{\mathbf{3}}{\mathbf{ℝ}}^{\mathbf{3}}$ .

Short answer

$\left[\begin{array}{cc}-2& -3\\ 0& 1\\ 1& 0\end{array}\right]$ is a matrix A such that$im\left(A\right)$ is the plane with normal vector$\left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]$ .

Step-by-step solution

Consider the parameters.

The image of a function consists of all the values the function takes in its target space. If f is a function from X to Y, then

$$\begin{gathered} image\left(f\right)=\left\{f\left(x:xinX\right)\right\} \\ =\left\{binY:b=f,forsomexinX\right\} \end{gathered}$$

Consider the system in terms of an equation,$x+3y+2z=0\Rightarrow x=-3y-2z$

When, $y=0\Rightarrow x=\left[\begin{array}{c}-2\\ 0\\ 1\end{array}\right]$

When, $z=0\Rightarrow x=\left[\begin{array}{c}-3\\ 1\\ 0\end{array}\right]$

Thus, $\left[\begin{array}{c}-2\\ 0\\ 1\end{array}\right],\left[\begin{array}{c}-3\\ 1\\ 0\end{array}\right]$are the required vectors.

Final answer.

$\left[\begin{array}{cc}-2& -3\\ 0& 1\\ 1& 0\end{array}\right]$ is a matrix A such that im(A) is the plane with normal vector$\left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]$ .