Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

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 $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$ .

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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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: 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.

Step 2: 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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Linear Algebra With Applications

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Short Answer

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 $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$ .

Step by Step Solution

Step 1: Linear combination

Step 2: Justification of answer

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Linear Algebra With Applications

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Short Answer

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] .

Step by Step Solution

Step 1: Linear combination

Step 2: Justification of answer

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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 $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$ .