Kernel of Matrix Homework Solution

[tangibleLime]

Homework Statement

Find the kernel of $$\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right)\$$

Homework Equations

The Attempt at a Solution

I know how to find the kernel of a matrix that has numbers in all of the columns, but I think that since this matrix has a zero vector in the first column it's throwing me off. I know the answer is the unit vector e1, but I don't know why.

I know that
$$imA = span(\left( \begin{array}{ccc} 1 \\ 0 \\ 0 \end{array} \right)\, \left( \begin{array}{ccc} 0 \\ 1 \\ 0 \end{array} \right)\)$$

And I know that the kernel is what makes Ax = 0. But I can't figure it out for some reason, while I can easily get the more complicated problems. Maybe it's too simple and I'm over-thinking it?

 

[micromass]

So, if the matrix had nonzero entries, how would you approach the problem then?? And what's different about this case??

 

[tangibleLime]

Yes, turns out I was overthinking it. I set it up like a normal problem and for some reason stopped because it looked strange. Then I continued with it and got the unit vector [1,0,0].