# Question about the Null Space for this Zero Matrix

1. Nov 15, 2004

### Theelectricchild

How can I determine the null space for the 2 x 6 zero matrix as precisely as I can?

Clearly N(A) = {x: Ax = 0, x in R^n},

So if A is this 2x6 matrix, wouldnt virtually any vector x that is in R^6 work?

This is supposed to be a "conceptual" problem, and I KNOW it cant be this easy for the bonus problem on the HW assignment!

Can anyone tell me what I am missing? THANKS A LOT!!!

2. Nov 15, 2004

### tiger_striped_cat

it's been a long time but check out

http://cnx.rice.edu/content/m10368/latest/

Basically you need to get A into row reduced echelon form. You have 2 equations, 6 unknowns (so you have 4 free parameters.)

x1 and x2 will be your piviots and your equation for x (your general vector which describes the set of all vectors in your null space is:

x = x3*a+x4*b+x5*c+x6*d

where a,b,c,d are your column vectors which give the coefficents of your x3,x4,x5,x6 when you solve for these variables. So for example, you should be able to get it down to where x1 is out of your system of equations and x2 is solved by all the other variables. So we can take the coefficient of this to also be zero. So we should have a vector in terms of the free variables only. Now solve this equation in terms of the remaining variables. X3=x3(x4,x5,x6), x4=x4(x3,x5,x6), and so on. So youll have zeros for x1,x2 in all a,b,c,d vectors, and 1s in the values for the variables you solve for, for "a" above, the 3rd element will be 1, because you're dealing with x3 here.

I'm too tired, and I'm sure there are some errors in this bad explaniation. But I hope it helps.

Last edited by a moderator: Apr 21, 2017
3. Nov 15, 2004

### Galileo

If A is the zero matrix (the matrix with all zero entries), then every vector x in R^6 will give Ax=0.
It's pretty trivial.
You could also use the rank equation.

4. Nov 15, 2004

### tiger_striped_cat

Did they really say zero matrix? That doesn't even seem like a problem.

5. Nov 15, 2004

### HallsofIvy

Staff Emeritus
Yes, this is a "conceptual" problem. What is the definition of "null space"??

6. Nov 15, 2004

### Divergent13

but did he not define it ?