Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q43E (page 160)

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 ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 3 \end{matrix}]$

Short Answer

Expert verified

If we have a plane $x_{1} + 2 x_{2} + x_{3} = 0$ with basis B consisting of vectors $[\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}]$ and $[\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}]$ and ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 3 \end{matrix}]$ then $\overset{鈫�}{x} = [\begin{matrix} 4 \\ - 3 \\ 2 \end{matrix}]$ .

Step by step solution

Mentioning the concept

Let $B = \{\overset{鈫�}{v}, \overset{鈫�}{w}\}$ be the basis of the given plane $x_{1} + 2 x_{2} + x_{3} = 0$
such that

$\overset{鈫�}{v} = [\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}]$ and $\overset{鈫�}{w} = [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}]$

Then any vector $\overset{鈫�}{x}$ in the plane can be uniquely written as:

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

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

i.e

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

$鈬� [\begin{matrix} 2 \\ - 3 \end{matrix}] = [\begin{matrix} c_{1} \\ c_{2} \end{matrix}]$

$鈬� c_{1} = 2 & 鈥� c_{2} = - 3 (2)$

Finding the vector x→

From equations (1) & (2), we have

_{$\begin{array}{l} \overset{鈫�}{x} = 2 [\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}] - 3 [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}] \\ 鈬� \overset{鈫�}{x} = [\begin{matrix} - 2 + 6 \\ 0 - 3 \\ 2 - 0 \end{matrix}] = [\begin{matrix} 4 \\ - 3 \\ 2 \end{matrix}] \\ 鈬� \overset{鈫�}{x} = [\begin{matrix} 4 \\ - 3 \\ 2 \end{matrix}] \end{array}$}

Final answer

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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 ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 3 \end{matrix}]$

Short Answer

Step by step solution

Mentioning the concept

Finding the vector x→

Final answer

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Consider the planex1+2x2+x3=0with basis B consisting of vectors [-101]and[-210]. If[x鈫�]B=[2-3]

Short Answer

Step by step solution

Mentioning the concept

Finding the vector x→

Final answer

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Decision Maths

Calculus

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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 ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 3 \end{matrix}]$