Question about the Null Space for this Zero Matrix

AI Thread Summary
The discussion centers on determining the null space of a 2x6 zero matrix, where it is established that any vector in R^6 will satisfy the equation Ax = 0. Participants clarify that the null space consists of all vectors in R^6 due to the zero matrix's properties. The conversation emphasizes the conceptual nature of the problem, noting that while it seems straightforward, understanding the definition of null space is crucial. Additionally, the process of row reducing the matrix and identifying free parameters is mentioned, but ultimately, the conclusion is that the null space is trivial. The discussion highlights the simplicity of the null space for a zero matrix, reinforcing that every vector in R^6 is included.
Theelectricchild
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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, wouldn't virtually any vector x that is in R^6 work?

This is supposed to be a "conceptual" problem, and I KNOW it can't be this easy for the bonus problem on the HW assignment!

Can anyone tell me what I am missing? THANKS A LOT!
 
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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.
 
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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.
 
Did they really say zero matrix? That doesn't even seem like a problem.
 
Theelectricchild said:
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, wouldn't virtually any vector x that is in R^6 work?

This is supposed to be a "conceptual" problem, and I KNOW it can't be this easy for the bonus problem on the HW assignment!

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

Yes, this is a "conceptual" problem. What is the definition of "null space"??
 
but did he not define it ?
 

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