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Q24E

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Linear Algebra With Applications

Found in: Page 120

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

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

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

$i m a g e (f) = \{f (x) : x i n X\} = \{b i n Y : b = f (x), for some x in X\}$

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

$[1 2 3] [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] = [14]$

Step 2: Final answer.

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

Most popular questions for Math Textbooks

Describe the images and kernels of the transformations in Exercises 23 through 25 geometrically.

23. Reflection about the line $y = \frac{x}{3} in ℝ^{2} ℝ^{2}$ .

Consider a subspace $V$ in $ℝ^{m}$ that is defined by homogeneous linear equations

$| \begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 m} x_{m} = 0 \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 m} x_{m} = 0 \\ ⋮ ⋮ ⋮ ⋮ \\ a_{n 1} x_{1} + a_{22} x_{2} + \dots + a_{nm} x_{m} = 0 \end{array} |$ .

What is the relationship between the dimension of $V$ and the quantity $m - n$

? State your answer as an inequality. Explain carefully.

In the accompanying figure, sketch the vector $\vec{x}$ with ${[\vec{x}]}_{؏} = [\begin{matrix} - 1 \\ 2 \end{matrix}]$ , where $؏$ is the basis of $ℝ^{2}$ consisting of the vectors $\vec{v}, \vec{w}$ .

Consider the matrices

$A = [\begin{matrix} 1 & 0 & 2 & 0 & 4 & 0 \\ 0 & 1 & 3 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] and B = [\begin{matrix} 1 & 0 & 2 & 0 & 0 & 4 \\ 0 & 1 & 3 & 0 & 0 & 5 \\ 0 & 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 0 & 1 & 7 \end{matrix}]$

Show that the kernels of the matrices A and B are different.

Find the basis of subspace of $ℝ^{4}$ that consists of all vectors perpendicular to both

$[\begin{matrix} 1 \\ 0 \\ - 1 \\ 1 \end{matrix}]$ and $[\begin{matrix} 0 \\ 1 \\ 2 \\ 3 \end{matrix}]$ .

See definition A.8 in the Appendix.

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