A Simple Linear Algebra Problem

[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 put...
Since the rank of the matrix is either 0, 1, or 2, I can conclude that the nullility is either 2, 1, or 0. So since there are 4 vectors in a 2 dimensional space, at most two are independent. Therefore the other rows must be dependent. '
Is this anywhere close? Thanks in advance.

 

[Tedjn]

What you've said is correct, although the last two sentences are what you really want. R² is 2-dimensional, so any 4 vectors must have dependencies. Of course, however, just from the question itself we can't tell what your professor would expect you to use; that would depend on what he's teaching in class.

 

[maherelharake]

So I don't have to add anything else to that? It makes sense, but it seems like that is more of a Chapter 2 explanation (we are in Chapter 5), and I was trying to explain it in a way that deals with our current material. Thanks

 

[Tedjn]

In the subject there would be theorems proved that say something to the effect that any set of n+1 vectors in an n dimensional vector space are never linearly independent. These considerations come up when you prove that every basis of a finite dimensional vector space has the same number of vectors; this number is defined to be the dimension. If you've covered this, then I don't see why you can't use the fact that R² is 2-dimensional.

On the other hand, with the current material, you might say that the column and row rank of a matrix are equivalent (which is what you hinted at), and that the row rank can be at most 2. By definition, row rank is the dimension of the rowspace; if it is less than or equal to 2, then 4 vectors certainly cannot be linearly independent. Essentially, these two arguments are almost equivalent, and they both rely on the concept of a unique dimension. I personally don't know another way to do this problem.

 

[maherelharake]

Ok thanks a lot. I will keep what I had, but add a little more explanation.