Question

The linear combination of $\overline{a}=\left[\begin{array}{c}1\\ 2\end{array}\right]and\overline{b}=\left[\begin{array}{c}0\\ 3\end{array}\right]$ which gives the vector $\left[\begin{array}{c}2\\ 2\end{array}\right]$ will be ___$\overline{a}$ +__ $\overline{b}$. (If any of the blanks is a fraction, enter the value to the nearest hundredth place).

Answer 1

A linear combination of $\overline{a}$ and $\overline{b}$ can be written as $c_1\overline{a}+c_2\overline{b}$
$So{c}_1\overline{a}+c_2\overline{b}=\left[\begin{array}{c}2\\ 2\end{array}\right]$
$or{c}_1\left[\begin{array}{c}1\\ 2\end{array}\right]$
$+c_2\left[\begin{array}{c}0\\ 3\end{array}\right]$ =$\left[\begin{array}{c}2\\ 2\end{array}\right]$
This gives
$(c_1\times 1)+(c_2\times 0)=2....\left(1\right)$
$(c_1\times 2)+(c_2\times 3)=2....\left(2\right)$
Multiply equation (1) by -2
$-2c_1+0=-4$
$\underset{–}{2c_1+3c_2=2}$ Now add these equations.
$3c_2=-2$
$c_2=\frac{-2}{3}=-1.66$
Also from equation (1)
$c_1=2$
So
$c_1=2$
$c_2=-1.66$
Linear combination will be
$2\overline{a}-1.66\overline{b}=\left[\begin{array}{c}1\\ 2\end{array}\right]$