Answer
${\bf{i}} = \frac{2}{{11}}{\bf{v}} + \frac{5}{{11}}{\bf{w}}$
Work Step by Step
We have ${\bf{v}} = \left( { - 2,5} \right)$, ${\bf{w}} = \left( {3, - 2} \right)$, and ${\bf{i}} = \left( {1,0} \right)$.
Write ${\bf{i}}$ as a linear combination $r{\bf{v}} + s{\bf{w}}$:
$\left( {1,0} \right) = r\left( { - 2,5} \right) + s\left( {3, - 2} \right)$
So, we have a system of equations:
$1 = - 2r + 3s$ ${\ \ }$ and ${\ \ }$ $0 = 5r - 2s$.
The second equation gives $s = \frac{5}{2}r$. Substituting it in the first equation gives $r = \frac{2}{{11}}$ and $s = \frac{5}{{11}}$. Thus, the linear combination of ${\bf{i}}$ in ${\bf{v}}$ and ${\bf{w}}$ is ${\bf{i}} = \frac{2}{{11}}{\bf{v}} + \frac{5}{{11}}{\bf{w}}$.
