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