Question

Give an example of a linear transformation whose image is the line spanned by $\left[\begin{array}{c}7\\ 6\\ 5\end{array}\right]$in${\mathbf{ℝ}}^{\mathbf{3}}$ .

Short answer

$\left[\begin{array}{ccc}7& 0& 0\\ 6& 0& 0\\ 5& 0& 0\end{array}\right]$is a linear transformation whose image is the line spanned by $\left[\begin{array}{c}7\\ 6\\ 5\end{array}\right]$ .

Step-by-step solution

Consider the parameters.

$7x+6y+5z=0$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

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

Consider the system in terms of an equation,

Thus, $\left[\begin{array}{ccc}7& 0& 0\\ 6& 0& 0\\ 5& 0& 0\end{array}\right]$ is a linear transformation.

Final answer.

$\left[\begin{array}{ccc}7& 0& 0\\ 6& 0& 0\\ 5& 0& 0\end{array}\right]$ is a linear transformation whose image is the line spanned by $\left[\begin{array}{c}7\\ 6\\ 5\end{array}\right]$.