Q46E Consider the plane x1+2x2+x3=0. ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q46E (page 160)

Consider the plane $x_{1} + 2 x_{2} + x_{3} = 0$ . Find a basis of this plane such that ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 1 \end{matrix}] for \overset{鈫�}{x} = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}]$ .

Short Answer

Expert verified

Let $\overset{鈫�}{v}$ be any vector in the plane $x_{1} + 2 x_{2} + x_{3} = 0$ , that is not parallel to $\overset{鈫�}{x}$ , then $\{\overset{鈫�}{v} (2 \overset{鈫�}{v} - \overset{鈫�}{x})\}$ is the basis of this plane.

Step by step solution

Finding B- coordinates of x→.

Let $B = \{\overset{鈫�}{v}, \overset{鈫�}{w}\}$ be the basis of the given plane $x_{1} + 2 x_{2} + x_{3} = 0$ . Then any $\overset{鈫�}{x}$ in the plane can be uniquely written as :

$\overset{鈫�}{x} = c_{1} \overset{鈫�}{v} + c_{2} \overset{鈫�}{w}$ (2)

where, $c_{1} & c_{2}$ are the B-coordinates of $\overset{鈫�}{x}$ .

i.e

${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} c_{1} \\ c_{2} \end{matrix}] 鈬� [\begin{matrix} 2 \\ - 1 \end{matrix}] = [\begin{matrix} c_{1} \\ c_{2} \end{matrix}] 鈬� c_{1} = 2 & c_{2} = - 1$

Therefore, from (1) & (2),

$[\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}] = 2 \overset{鈫�}{v} - \overset{鈫�}{w}$

Finding the basis B of the plane x1+2x2+x3=0.

Let $\overset{鈫�}{v} = {(v_{1}, v_{2}, v_{3})}^{T} & \overset{鈫�}{w} = {(w_{1}, w_{2}, w_{3})}^{T}$

Then,

$[\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}] = 2 {(v_{1}, v_{2}, v_{3})}^{T} - {(w_{1}, w_{2}, w_{3})}^{T} 鈬� 2 v_{1} - w_{1} = 1 2 v_{2} - w_{2} = - 1 2 v_{3} - w_{3} = 1$

Therefore,

${(v_{1}, v_{2}, v_{3})}^{T} = 2 {(w_{1}, w_{2}, w_{3})}^{T} = {(1, - 1, 1)}^{T}$

Final answer

If $\overset{鈫�}{v}$ be any vector in the plane $x_{1} + 2 x_{2} + x_{3} = 0$ , that is not parallel to $\overset{鈫�}{x}$ , then $B = \{\overset{鈫�}{v} (2 \overset{鈫�}{v} - \overset{鈫�}{x})\}$ is the basis of this plane.

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91影视

Consider the plane $x_{1} + 2 x_{2} + x_{3} = 0$ . Find a basis of this plane such that ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 1 \end{matrix}] for \overset{鈫�}{x} = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}]$ .

Short Answer

Step by step solution

Finding B- coordinates of x→.

Finding the basis B of the plane x1+2x2+x3=0.

Final answer

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91影视

Consider the plane x1+2x2+x3=0. Find a basis of this plane such that [x鈫�]B=[2-1]forx鈫�=[1-11].

Short Answer

Step by step solution

Finding B- coordinates of x→.

Finding the basis B of the plane x1+2x2+x3=0.

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Pure Maths

Study anywhere. Anytime. Across all devices.

Consider the plane $x_{1} + 2 x_{2} + x_{3} = 0$ . Find a basis of this plane such that ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 1 \end{matrix}] for \overset{鈫�}{x} = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}]$ .