Q20E Determine whether the statements... [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 1: Q20E (page 39)

Determine whether the statements that follow are true or false, and justify your answer.
20:Vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is a linear combination of $[\begin{matrix} 4 \\ 5 \\ 6 \end{matrix}]$ and $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$ .

Short Answer

Expert verified

It is true that Vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is a linear combination of $[\begin{matrix} 4 \\ 5 \\ 6 \end{matrix}]$ and $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$ .

Step by step solution

Linear combination

A vector is said to be a linear combination of other vectors if it can be written as in the linear combination of other vectors.

Justification of answer

Let vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is a linear combination of $[\begin{matrix} 4 \\ 5 \\ 6 \end{matrix}]$ and $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$ .

Then it will imply

$[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] = x [\begin{matrix} 4 \\ 5 \\ 6 \end{matrix}] + y [\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}] [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] = [\begin{matrix} 4 x + 7 y \\ 5 x + 8 y \\ 6 x + 9 y \end{matrix}]$

Now equating the matrix we will get

$4 x + 7 y = 1 5 x + 8 y = 2 6 x + 9 y = 3$

On solving above equations, we will get

$x = 2, y = - 1$

Hence, it is true that Vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is a linear combination of $[\begin{matrix} 4 \\ 5 \\ 6 \end{matrix}]$ and $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$

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91影视

Determine whether the statements that follow are true or false, and justify your answer.
20:Vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is a linear combination of $[\begin{matrix} 4 \\ 5 \\ 6 \end{matrix}]$ and $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$ .

Short Answer

Step by step solution

Linear combination

Justification of answer

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91影视

Determine whether the statements that follow are true or false, and justify your answer.20:Vector [123] is a linear combination of [456]and[789] .

Short Answer

Step by step solution

Linear combination

Justification of answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Calculus

Applied Mathematics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Determine whether the statements that follow are true or false, and justify your answer.
20:Vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is a linear combination of $[\begin{matrix} 4 \\ 5 \\ 6 \end{matrix}]$ and $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$ .