null space for a matrix

Ok, so here is your matrix:

A=⎛⎝⎜⎜000021000410−5001⎞⎠⎟⎟

You want to know what the nullspace is of the matrix A. So this means you have an equation of the form.

Ax⃗ =0⃗ 

which will look like (sorry LaTeX wasn't playing nice for me here, I'll just write it out):
[0 2 0 -5][x1] = [0]
[0 1 4 0][x2] = [0]
[0 0 1 0][x3] = [0]
[0 0 0 1][x4] = [0]



So to solve this you could:
Use gaussian-jordon elimination and reduce it to:

⎛⎝⎜⎜0000010000100001⎞⎠⎟⎟

as Hammie pointed out.

So this leaves you with:

x1=α

x2=0

x3=0

x4=0

Which is a matrix of the form:

(α,0,0,0)T

which typically you want to factor out the variables to yield:

x⃗ =α(1,0,0,0)T

Now thinking of this in the algebraic sense, the solution to the system is:
(going back to the a,b,c,d notation:

a=α

b=0

c=0

d=0

Which, just as Hammie said. b,c,d are for the trivial solution while a could be anything.

I think an easy general method for solving for the null space is to:
1) reduce it as much as possible:
ex) (1 0 0, 0 1 0, 0 0 0)^T type of matrix

2) any row that is all 0's then set it equal to some variable.
3) write down a column vector describing the solutions, and factor out the variables.