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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q24E (page 120)

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

Short Answer

Expert verified

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

Step by step solution

01

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,

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

02

Final answer.

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

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Most popular questions from this chapter

Consider the plane $x_{1} + 2 x_{2} + x_{3} = 0$ with basis B consisting of vectors $[\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}] and [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}]$ . If ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 3 \end{matrix}]$

Find the basis of subspace of ${鈩�}^{4}$ 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.

Consider three linearly independent vectors ${\overset{鈫�}{V}}_{1}, {\overset{鈫�}{V}}_{2}, {\overset{鈫�}{V}}_{3}$ in ${鈩�}^{4}$ .Find $r r e f [\begin{matrix} | & | & | \\ \overset{鈫�}{v_{1}} & \overset{鈫�}{v_{2}} & \overset{鈫�}{v_{3}} \\ | & | & | \end{matrix}]$

Prove Theorem 3.3.4d: If 鈥榤鈥� vectors spans an m-dimensional space, they form a basis of the space.

Determine whether the following vectors form a basis of ${鈩�}^{4}$ ; $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ - 1 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 4 \\ 8 \end{matrix}], [\begin{matrix} 1 \\ - 2 \\ 4 \\ - 8 \end{matrix}]$ .

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