Null space for Matrix with similar columns

• Nov 17th 2011, 01:47 PM
nkhan
Null space for Matrix with similar columns
Hi,

I just read that the null space for the following matrix

[1 0 1;
5 4 9;
2 4 6]

is the line of all points x = c, y = c, z = -c.

I was wondering what the null space for the matrix with three similar columns would be e.g.

[1 1 1;
0 0 0;
0 0 0]

(c, -c, 0), (-c, 2c, -c), etc are all part of the null space for this matrix but they don't form a line nor a plane.

Any help would be much appreciated.

Regards
• Nov 17th 2011, 02:08 PM
Deveno
Re: Null space for Matrix with similar columns
the matrix:

$\begin{bmatrix}1&1&1\\0&0&0\\0&0&0\end{bmatrix}$

has rank 1, so it has null space of dimension 2. it's clear that (x,y,z) is in this nullspace iff x+y+z = 0, that is if (x,y,z) is of the form: (s,t,-s-t),

which is the plane: s(1,0,-1) + t(0,1,-1). a basis is {(1,0,-1),(0,1,-1)}.

both of your example points lie in this plane. simply take s = c, t = 0, for the first one, and s = -c, t = 2c for the second.