How to test if vectors span a space?

[ramparts]

Hey folks - I left my linear book at home for the summer and am having trouble finding this on the Google, so I'm hoping you'll be good for an easy question :) I have a set of vectors that I think spans a vector space, but I've forgotten how to test it. I can make a set of 5 simultaneous equations (it's a 5-dimensional space) but that seems a bit much. How do I do this again?

I'd also appreciate a reminder about how to test for linear independence, though it seems with this particular set of vectors that part is pretty trivial.

Thanks!

 

[slider142]

If you have 5 vectors that are linearly independent, then they span a 5-dimensional vector space. (n-1) vectors can never span an n-dimensional space. One test for linear independence, besides using the definition, is to find the determinant of the matrix of the n vectors in question. If it is 0, the list of vectors is linearly dependent.

 

[ramparts]

I didn't know it was that simple, thanks! If the determinant of the n-by-n matrix is non-zero, presumably that means the vectors are linearly independent?

 

[ramparts]

Sorry, I left out something obviously important - I'm looking to see if these vectors span R^5.

 

[jambaugh]

Note that you can still have a set of linearly dependent vectors spanning a given space (you'd have more vectors than the dimension of that space). The sufficient condition is to express each of the space's basis elements as linear combinations of the set of vectors you are considering. If you have "too many" vectors then there will be more than one way to do this.

Note that you may for example be dealing with an infinite dimensional space. For example does the set of polynomials in x span the space of analytic functions (of x) on the real number line? Answer is yes, proof involves showing every analytic function has a power series expansion i.e. is a linear combination of monomials.

 

[ramparts]

Yep - I was looking for a basis set, but it was pretty trivial that it was linearly independent so the main thing was to find span. Just had to sharpen my linear algebra, I put the vectors in a matrix and row reduced, and yeah, it spans.

It was actually a mildly interesting problem, I was curious if full-on Planck units (setting G, h-bar, c, e_0 and k to 1) uniquely covered any combination of base units (they do).

 

[diazona]

Oh, that 5-dimensional vector space On that note, this thread (link) may interest you.