Q45E Consider the plane 2x1-3x2+4x3=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: Q45E (page 160)

Consider the plane $2 x_{1} - 3 x_{2} + 4 x_{3} = 0$ . Find a basis $B$ of this plane such that ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ 3 \end{matrix}]$ for $\overset{鈫�}{x} = [\begin{matrix} 2 \\ 0 \\ - 1 \end{matrix}]$ .

Short Answer

Expert verified

Thus, the basis is $B = [\begin{matrix} 3 \\ 2 \\ 0 \end{matrix}], \frac{1}{3} [\begin{matrix} - 4 \\ - 4 \\ - 1 \end{matrix}]$ .

Step by step solution

Find a basis B plane such that xΒ→=23 for x→=20-1.

Consider a plane $2 x_{1} - 3 x_{2} + 4 x_{3} = 0$ .

And the vector is given as;

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

With the coordinate as follows:

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

Let $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ are basis for plane.

As ${[\overset{鈫�}{x}]}_{螔} = [\begin{matrix} 2 \\ 3 \end{matrix}]$ therefore role="math" localid="1664267022537" $\overset{鈫�}{x} = 2 \overset{鈫�}{v} + 3 \overset{鈫�}{w}$

Now, solve for $\overset{鈫�}{w}$ as follows:

$x = 2 \overset{鈫�}{v} + 3 \overset{鈫�}{w} 3 \overset{鈫�}{w} = \overset{鈫�}{x} - 2 \overset{鈫�}{v} \overset{鈫�}{w} = \frac{1}{3} (\overset{鈫�}{x} - 2 \overset{鈫�}{v})$

This means that the vectors $\overset{鈫�}{v} and \frac{1}{3} (\overset{鈫�}{x} - 2 \overset{鈫�}{v})$ (here vector $\overset{鈫�}{v}$ is not parallel to $\overset{鈫�}{x}$ ) form a basis for plane.

Example for the given plane.

For example let $\overset{鈫�}{v} = [\begin{matrix} 3 \\ 2 \\ 0 \end{matrix}]$ be any solution to $2 x_{1} - 3 x_{2} + 4 x_{3} = 0$ .

Calculate the other vectors as follows;

$\frac{1}{3} (\overset{鈫�}{x} - 2 \overset{鈫�}{v}) = \frac{1}{3} ([\begin{matrix} 2 \\ 0 \\ - 1 \end{matrix}] - 2 [\begin{matrix} 3 \\ 2 \\ 0 \end{matrix}]) = \frac{1}{3} [\begin{matrix} - 4 \\ - 4 \\ - 1 \end{matrix}]$

Therefore, the basis is $B = [\begin{matrix} 3 \\ 2 \\ 0 \end{matrix}], \frac{1}{3} [\begin{matrix} - 4 \\ - 4 \\ - 1 \end{matrix}]$ .

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91影视

Consider the plane $2 x_{1} - 3 x_{2} + 4 x_{3} = 0$ . Find a basis $B$ of this plane such that ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ 3 \end{matrix}]$ for $\overset{鈫�}{x} = [\begin{matrix} 2 \\ 0 \\ - 1 \end{matrix}]$ .

Short Answer

Step by step solution

Find a basis B plane such that xΒ→=23 for x→=20-1.

Example for the given plane.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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91影视

Consider the plane 2x1-3x2+4x3=0. Find a basis Bof this plane such thatx鈫�B=23 for x鈫�=20-1.

Short Answer

Step by step solution

Find a basis B plane such that xΒ→=23 for x→=20-1.

Example for the given plane.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Decision Maths

Calculus

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Consider the plane $2 x_{1} - 3 x_{2} + 4 x_{3} = 0$ . Find a basis $B$ of this plane such that ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ 3 \end{matrix}]$ for $\overset{鈫�}{x} = [\begin{matrix} 2 \\ 0 \\ - 1 \end{matrix}]$ .