What is the Dimension of the Null Space for Matrix A?

[snoggerT]

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

| 1 3|
|-2 -6|

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.

 

[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?

 

[snoggerT]

will using this work...

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

using that formula gave me the correct answer.

 

[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.

 

[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.

 

[snoggerT]

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

 

[HallsofIvy]

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?