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Found in: Page 120

### Linear Algebra With Applications

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

# Describe the images and kernels of the transformations in Exercises 23 through 25 geometrically. 24. Orthogonal projection onto the plane ${\mathbit{x}}{\mathbf{+}}{\mathbf{2}}{\mathbit{y}}{\mathbf{+}}{\mathbf{3}}{\mathbit{z}}{\mathbf{=}}{\mathbf{0}}$ in ${{\mathbf{ℝ}}}^{3}$ .

The orthogonal projection is [14] , image of the kernel is, $\left\{\left(x,y,z\right):x+2y+3z=0\right\}$ .

See the step by step solution

## Step 1: 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

$image\left(f\right)=\left\{f\left(x\right):xinX\right\}\phantom{\rule{0ex}{0ex}}=\left\{binY:b=f\left(x\right),\mathrm{for}\mathrm{some}x\mathrm{in}X\right\}$

The orthogonal projection of $x+2y+3z=0$ is,

$\left[123\right]\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]=\left[14\right]$

## Step 2: Final answer.

The orthogonal projection is $\left[14\right]$ , image of the kernel is, $\left\{\left(x,y,z\right):x+2y+3z=0\right\}$ .