for the 
matrix 
.
Column space and nullspace
Here's the picture of for the matrix .
You see that 
so the singular values are
s1 = 3 and s2 = 0.
What's the effect of the zero singular value 
?
First, you can check that every vector 
is sent to a multiple of 
.
The red line in the graphic shows the multiples of 
.
This line is also known as the column space of 
.
The matrix 
sends every vector 
to some point on the red line.
One thing this tells you is that the equation 
can have a solution only if 
is on the red line. (This is why you want to know about the column space of a matrix.)
A second effect of 
is that changing the amount of 
in 
does not affect 
.
Set the 
part of 
to zero and sweep out the blue line by changing the amount of 
. The blue line is the nullspace of 
.
(Q27) Where does matrix 
sends all points on the blue line?
Set the 
part of 
to 
and sweep out the green line by changing the amount of 
.
(Q28) Where does matrix 
sends all points on the green line?
You can check that for any line parallel to the nullspace, 
sends all points on the line to the same place. (This is the reason you want to know about the nullspace of a matrix.)
Copyright © 2007 Todd Will
Created by
Mathematica (April 14, 2007)