Plotting null and column space of C

Question

I can't figure out how to properly plot the null and column space of a certain matrix

$C = \begin{pmatrix} 6 & -12 & 6 \\ 12 & -36 & 18 \\ 18 & -36 & 18 \end{pmatrix} $.

For the null space I reduce C to $C = \begin{pmatrix} 1& 0 & 0 \\ 0 & 1 & -1/2 \\ 0 & 0 & 0 \end{pmatrix} $ and try to us the matlab function plot to plot the nullspace.

k = -10:10;
x_3 = k;
x_1 = 0*k;
x_2 = 1/2 * x_3;
plot3(x_1,x_2,x_3)
xlabel('x1')
zlabel('x3')
ylabel('x2')
title('Null(C)');

Yields this plot. I am not sure though if it is correct, and have little clue how to begin on plotting Col(C).

Answer 1

The nullspace is a line in the $x_2−x_3$ plane with slope $\mathrm{d}y/\mathrm{d}z=1/2$ and that passes through the origin (all subspaces must include the origin). So all you have to do is plot this line over some finite range and it looks like you have done so to me.