question checknig if a set of vectors span a vector space(REFRASED)
Author
my question is this...
If I have a set of n vectors in Rn, and that set is linearly dependant, is it possible for that set to span Rn?
I hope that is a clearer question :)
[Edited by - donjonson on July 17, 2005 3:42:11 PM]
Yes, at least if I get what you're asking, check: http://mathworld.wolfram.com/VectorSpaceBasis.html.
If thats not quite what you mean could you rephrase it again :)
If thats not quite what you mean could you rephrase it again :)
Quote:Original post by donjonson
If I have a set of n vectors in Rn, and that set is linearly dependant, is it possible for that set to span Rn?
No, in order to span Rn, you need at least n linearly independent vectors. I think, however, you probably meant independent and not dependant. In that case, I'm not sure what the conditions are...being independent might be sufficient.
[Edited by - Promit on July 17, 2005 6:32:06 PM]
Short answer: no.
Long answer: A linearly dependent set of n vectors can not span Rn. If you have a linearly dependent set of vectors u1, u2,... un that span a space Un then you could just remove the vectors that can be expressed as a linear combination of the others until you have a linearly independent subset u1, u2,... uk (k<n) which stills spans Un. Problem is, you need at least n vectors to span Rn, so you're screwed.
Edit: Never saw your post right above mine, I was answering your original question, sorry if you got confused.
[Edited by - GameCat on July 18, 2005 6:08:03 PM]
Long answer: A linearly dependent set of n vectors can not span Rn. If you have a linearly dependent set of vectors u1, u2,... un that span a space Un then you could just remove the vectors that can be expressed as a linear combination of the others until you have a linearly independent subset u1, u2,... uk (k<n) which stills spans Un. Problem is, you need at least n vectors to span Rn, so you're screwed.
Edit: Never saw your post right above mine, I was answering your original question, sorry if you got confused.
[Edited by - GameCat on July 18, 2005 6:08:03 PM]
Quote:Original post by donjonson
so in other words, any set of n vectors span Rn then it is also a basis for Rn?
Yes. And furthermore, you can use Graham-Schmidt orthonormalization to come up with an orthonormal basis for Rn.
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