Column space and nullspace

Here's the picture of for the matrix .

You see that so the singular values are
s_{1} = 3 and s_{2} = 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.)

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Copyright © 2007 Todd Will

Created by
Mathematica (April 14, 2007)