Thread: Span and consistency.

Hello there,

I've been given a question that asks me to determine whether a given vector is in the span of a set of vectors. I understand the concept of span and how to tell whether a vector is in the span, but there is one small thing I'm not quite certain of.

I know that to find out whether a vector is in the span, you row reduce the matrix containing the vectors in the set with the vector you're trying to test in the last column. If the row reduced outcome is consistent, then the vector is in the span. Of course I could have read the chapter wrong and not been doing it right, so if that's the case, please let me know.

My question is, if your outcome is a perfectly row reduced matrix, with 4 pivot columns in a 4x4 matrix for example, does this still count as an inconsistent system because of the last row?

Hopefully I explained that right, but let me know if I didn't. Thanks in advance for the help!

Your question is not clear because you have not said what the dimemsion of your space is. Also when you say "the matrix containing the vectors", containing them how? As columns is most common but you should say that.


If, for example, you have two vectors in your set and it is in three dimensions, your "augmented matrix" will be three by three, not 4 by 4. I think it would be better to think about what "span" means, which you say you know. A vector, x, is in the "span" of the set of vectors {u, v, w}, if and only if there exist numbers a, b, c such that x= au+ bv+ cw. Of course, it you write out the components, you will have a system of equations you could then put into an "augmented matrix" which is essentially what you are doing. Saying that you get a 4 by 4 matrix implies that you have a set of 3 vectors each having 4 components. Is that correct?