A Couple Linear Algebra Problems

[maherelharake]

Homework Statement

If A is a 4x2 matrix, explain why the rows of A must be linearly dependent.

Homework Equations

The Attempt at a Solution

I said that nullility(A)=n-rank(A). This tells you that nullility(A) could be equal to 0,1, or 2. Since there are 4 vectors, at least two of them have to be linearly dependent. Is this correct?

Homework Statement

If A is a 4x2 matrix, what are the possible values of nullity(A)?

Homework Equations

The Attempt at a Solution

I said that the nullility(A) is 2-2 or 2-1 or 2-0, causing the answer to be 0, 1 or 2.

I just don't know if I thought about these correctly. Thanks in advance.

 

[Dick]

I'm not totally clear what you are thinking. WHY is rank A=0,1,2? I'm not saying it's wrong, in fact, it's completely correct. Just why?

 

[maherelharake]

I just visualized a 4x2 matrix and figured that the number of nonzero rows had to be either 0, 1 or 2. Is that the right way to think of it?

 

[Dick]

Well, ALL of the rows might be nonzero. I'm guessing you are visualizing row reduction of the matrix, right? Sure, the rows are four vectors in a two dimensional space. At most two of them are linearly independent. So yes, rank=0,1 or 2.

 

[maherelharake]

Yes I am visualizing row reduction, sorry I forgot to say that. And that is the proof of why the rows of A must be dependent, right? Since at most two are independent?

 

[Dick]

Sure. That works for me.

 

[maherelharake]

Ok thanks. Is the second part correct too? The part about nullility? They seem to go hand-in-hand.

 

[Gear300]

The maximum rank A can have is 2. A^T has the same rank as A, so each row vector is linearly dependent on at least one other row vector. The nullity is the difference between the number of columns of A and the rank of A...so it looks like you got things right.

 

[Dick]

Ok thanks. Is the second part correct too? The part about nullility? They seem to go hand-in-hand.

Yeah. You are right about that part. 4=rank+nullity. If rank=0,1,2, then nullity=0,1,2.

 

[maherelharake]

Ok great. It seems like everything I had was correct, but I just needed to make sure. I might have to post another thread tomorrow too if some more problems come up. Thanks for your help guys.