Finding kernel?

If anyone could help me with the following problem, it would be greatly appreciated!

Consider the linear transformation L: M_22 -> R^4 defined by
\(\displaystyle L\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) = (-10a - c + 2d, 5a + b - d, -2c + d, 9a + 17b + 2d).\)

Compute ker(L).

Okay, so far I've put the coefficients into a matrix and row reduced the matrix to get

1st row: 1, 0, 0, 0
2nd row: 0, 1, 0, 0
3rd row: 0, 0, 1, 0
4th row: 0, 0, 0, 1


But I'm unsure of where to go from there.

Well that means that you can have only a trivial solution.
So what is the kernel?

Does this mean the kernel is simply {0}?

Yes but let's write correctly:
\(\displaystyle \L\left\{ {\left( {\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}} \right)} \right\}.\)