Dimension of Null Space

1. Mar 3, 2008

snoggerT

Find the Dimension of the null space of the given matrix A:

| 1 3|
|-2 -6|

3. The attempt at a solution
I honestly don't know how to work this at all. I think I'm confused as to what Null Space actually is, so that's making this a difficult problem to understand. please help.

2. Mar 3, 2008

Dick

A vector x is in the null space if Ax=0. If you write x as the column vector (u,v) then you want to solve the linear system u+3v=0 and -2u-6v=0. You must have done something like that before, right? How do you do it?

3. Mar 3, 2008

snoggerT

will using this work...

# of variables - Rank(A)= dim(null space A) ?

using that formula gave me the correct answer.

4. Mar 3, 2008

Dick

Sure. If you found the rank of A=1 then the dimension of the null space is 2-1. You could also have solved the linear system to find the answer can be expressed in terms of one parameter. That also means the dimension of the null space is one.

5. Mar 3, 2008

frasifrasi

the dimension is just how many vectors you get out of the kernel.

if there are 2 variables and the rank is 1 (one leading 1) , there must be 1 free variable, which means the dim will be 1 and there will only be 1 vector for the kernel.

6. Mar 3, 2008

snoggerT

that makes a lot more sense to me now. thanks.

7. Mar 4, 2008

HallsofIvy

Staff Emeritus
I really dislike using formulas like that when it can be done straight from the definitions:

v is in the null space of A if and only if

$$\left(\begin{array}{cc}1 & 3 \\-2 & -6 \end{array}\right)\left(\begin{array}{c}x \\ y\end{array}\right)= \left(\begin{array}{c}0 \\ 0\end{array}\right)$$
which is the same as saying x+ 3y= 0 and -2x- 6y= 0. What (x, y) satisfy both of those equations?