Question

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

Short answer

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

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 $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$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

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

Now equating the matrix we will get

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

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