What is the nullspace of a 3x3 complex matrix?

[math2010]

Homework Statement

I have the 3x3 matrix C=(1,-1,1; 2,0,1+i; 0,1+i,-1) and I want to find its nullspace (a set of vectors that span that subspace).

The Attempt at a Solution

So first I have reduced the matrix to row echelon form and I got this matrix:
(1,-1,1; 0,1,-0.5+0.5i; 0,0,0)

How do I read off from this the nullspace of this matrix? What is a basis for this nullspace?

By "i" I mean imaginary since this is a complex matrix.

 

[Mark44]

I ended up with a different row-reduced matrix, with no rows of zeroes.

 

[math2010]

Are you sure? Because I used Mathematica to check the reduced row echelon form of this matrix, and it seems the rref has a row of zeros!

Also, does the set containing (-1,0.5+0.5i,1) and (1,0,0) span the subspace?

 

[math2010]

I tried row-reducing it again using Matlab and I still got a zero row:

1 0 0.5 + 0.5i
0 1 -0.5 + 0.5i
0 0 0

 

[vela]

That reduced matrix corresponds to equations

$$x+(0.5+0.5 i)z = 0$$
$$y+(-0.5+0.5 i)z = 0$$

Solving for the other variables in terms of z, you get a solution of

$$\begin{pmatrix}x\\y\\z\end{pmatrix}=z\begin{pmatrix}-0.5-0.5i\\0.5-0.5i\\1\end{pmatrix}$$

The vector multiplying the z on the RHS is a basis of the nullspace.

 

[Mark44]

I tried row-reducing it again using Matlab and I still got a zero row:

1 0 0.5 + 0.5i
0 1 -0.5 + 0.5i
0 0 0

I agree with your result now.