Solving systems of linear equations in the complex field. I need to show that two systems of linear equations are equivalent, however this is over the complex field. 
How would I solve this?
One of the systems is:
$$\left\{\begin{align*}
&2x_1+(-1+i)x_2+x_4=0\\
&3x_2-3ix_3+5x_4=0\;.
\end{align*}\right.$$ 
Yet no matter how many ways I look at it, I can't find a way to reduce it enough in the matrix that it's practical to solve. Is there another way of showing that two systems are equivalent? 

You solve it in the exact same way as in the real case. That is, since it's underdetermined, you need to pick two variables you'll leave undetermined (let's say $x_3$ and $x_4$), and form the matrix equation
$$\left[\begin{array}{cc}2 & -1+i \\0 & 3\end{array}\right]\left[\begin{array}{c}x_1\\x_2\end{array}\right]=\left[\begin{array}{c}-x_4\\3ix_3-5x_4\end{array}\right]$$
and then invert the matrix to get
$$\left[\begin{array}{c}x_1\\x_2\end{array}\right]=\frac{1}{6}\left[\begin{array}{cc}3 & 1-i \\0 & 2\end{array}\right]\left[\begin{array}{c}-x_4\\3ix_3-5x_4\end{array}\right] = \frac{1}{6}\left[\begin{array}{c}(-3+3i)x_3+(-8+5i)x_4\\6ix_3-10x_4\end{array}\right]$$
Now you can do the same thing to your other system and verify that you get the same expressions for $(x_1,x_2)$.
