Answer
${\bf{u}} = \frac{{11}}{{18}}{\bf{v}} + \frac{5}{{18}}{\bf{w}}$,
where ${\bf{v}} = \left( {1,4} \right)$ and ${\bf{w}} = \left( {5,2} \right)$.
Work Step by Step
We express ${\bf{u}}$ as a linear combination of ${\bf{v}}$ and ${\bf{w}}$:
${\bf{u}} = r{\bf{v}} + s{\bf{w}}$
$\left( {2,3} \right) = r\left( {1,4} \right) + s\left( {5,2} \right)$
Now we solve the system of equations:
$2 = r + 5s$ ${\ \ }$ and ${\ \ }$ $3=4r+2s$.
So, we have $r=2-5s$. Substituting it in the equation $3=4r+2s$ gives
$3=4(2-5s)+2s$.
The solutions are $s = \frac{5}{{18}}$ and $r = \frac{{11}}{{18}}$.
Thus, ${\bf{u}} = \frac{{11}}{{18}}{\bf{v}} + \frac{5}{{18}}{\bf{w}}$.
