Answer
$u=\left(\dfrac{1}{2},0\right)=\dfrac{1}{2}i$
$v=\left(\dfrac{1}{8},-\dfrac{1}{4}\right)=\dfrac{1}{8}i-\dfrac{1}{4}j$
Work Step by Step
We are given the system of equations:
$\begin{cases}
2u=i\\
u-4v=j
\end{cases}$
Let's note:
$u=(u_1,u_2)$
$v=(v_1,v_2)$
Rewrite the system:
$\begin{cases}
2(u_1,u_2)=(1,0)\\
(u_1,u_2)-4(v_1,v_2)=(0,1)
\end{cases}$
$\begin{cases}
(2u_1,2u_2)=(1,0)\\
(u_1,u_2)-(4v_1,4v_2)=(0,1)
\end{cases}$
$\begin{cases}
2u_1=1\\
2u_2=0\\
(u_1-4v_1,u_2-4v_2)=(0,1)
\end{cases}$
$\begin{cases}
u_1=\dfrac{1}{2}\\
u_2=0\\
u_1-4v_1=0\\
u_2-4v_2=1
\end{cases}$
$u_1-4v_1=0$
$\dfrac{1}{2}-4v_1=0$
$\dfrac{1}{2}=4v_1$
$v_1=\dfrac{1}{8}$
$u_2-4v_2=1$
$0-4v_2=1$
$4v_2=-1$
$v_2=-\dfrac{1}{4}$
The solution is:
$u=(u_1,u_2)=\left(\dfrac{1}{2},0\right)=\dfrac{1}{2}i$
$v=(v_1,v_2)=\left(\dfrac{1}{8},-\dfrac{1}{4}\right)=\dfrac{1}{8}i-\dfrac{1}{4}j$
