Answer
$(4,-8)=-2u-6v$
Work Step by Step
We are given the vectors:
$w=(4,-8)$.
$u=(1,1)$
$v=(-1,1)$
We have:
$w=(4,-8)=c_1u+c_2v$
$(4,-8)=c_1(1,1)+c_2(-1,1)$
$(4,-8)=(c_1,c_1)+(-c_2,c_2)$
$(4,-8)=(c_1-c_2,c_1+c_2)$
Determine $c_1$ and $c_2$:
$\begin{cases}
c_1-c_2=4\\
c_1+c_2=-8
\end{cases}$
$c_1-c_2+c_1+c_2=4-8$
$2c_1=-4$
$c_1=-2$
$c_2=-8-c_1=-8-(-2)=-6$
The vector can be written:
$(4,-8)=-2u-6v$
