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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.

## Step 2: Justification of answer

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 $\left[\begin{array}{c}7\\ 8\\ 9\end{array}\right]$