Column space and nullspace

Here's the picture of xA x for the 2  2 matrix A = (v    v ) (s      ) (u )       1    2    1   0     1                      s    u                0     2    2.

You see that A(u_1 + u_2) = 3v_1 + 0v_2 so the singular values are s1 = 3 and  s2 = 0.

What's the effect of the zero singular value s_2 = 0?

First, you can check that every vector x is sent to a multiple of v_1.  
The red line in the graphic shows the multiples of v_1.  
This line is also known as the column space of A.
The matrix A sends every vector x to some point on the red line.

One thing this tells you is that the equation A x = y can have a solution only if y is on the red line.  (This is why you want to know about the column space of a matrix.)

A second effect of s_2 = 0 is that changing the amount of u_2 in x does not affect  A x .   

Set the u_1 part of x to zero and sweep out the blue line by changing the amount of u_2.  The blue line is the nullspace of A.

(Q27) Where does matrix A sends all points on the blue line?

Set the u_1 part of x to 2 u_1 and sweep out the green line by changing the amount of u_2.  

(Q28) Where does matrix A sends all points on the green line?

You can check that for any line parallel to the nullspace, A sends all points on the line to the same place.  (This is the reason you want to know about the nullspace of a matrix.)


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Copyright 2007 Todd Will
Created by Mathematica  (April 14, 2007)