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Q20E

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Found in: Page 39

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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

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

See the step by step solution

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.

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

Then it will imply

$\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]=x\left[\begin{array}{c}4\\ 5\\ 6\end{array}\right]+y\left[\begin{array}{c}7\\ 8\\ 9\end{array}\right]\phantom{\rule{0ex}{0ex}}\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]=\left[\begin{array}{c}4x+7y\\ 5x+8y\\ 6x+9y\end{array}\right]$

Now equating the matrix we will get

$4x+7y=1\phantom{\rule{0ex}{0ex}}5x+8y=2\phantom{\rule{0ex}{0ex}}6x+9y=3\phantom{\rule{0ex}{0ex}}$

On solving above equations, we will get

$x=2,y=-1$

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