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Found in: Page 120

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# 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 ${{\mathbf{ℝ}}}^{{\mathbf{3}}}$ .

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

See the step by step solution

## Step 1: 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

$image\left(f\right)=\left\{f\left(x\right):xinX\right\}\phantom{\rule{0ex}{0ex}}=\left\{binY:b=f\left(x\right),forsomexinX\right\}$

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.

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