I remember reading that if you have a set of vectors that if one of the vectors is a lin combination of the vectors before it then the set is lin dependent. What I'm confused about is does that vector have to be lin combination of ALL of the previous vectors or just at least one for the set to be lin dependent?

Question

zarkon · Answer

a combination of one or more


though if you can write it is a l.c. of one of the vectors you can write it is a linear combination of all the vectors

anonymous · Answer

I think should be all the combinations.

anonymous · Answer

Great so if you want to get the basis of the set of vectors you get rid of the linear combination right?

zarkon · Answer

you get rid off was many vectors as needed to make them independent (but no more)

zarkon · Answer

*of

anonymous · Answer

Ok let's say we have a a set of 3 vectors. We can't determine if it's lin dependent by inspection yet when we reduce it to echelon form we find a free variable so we only have two pivot points but this was done through row reduction using all the rows meaning we added or subtracted row 1 to row 2 and then used row 2 to make the third row a zero row. 

At that point if we want to have the basis of the set we have to get rid of a vector. Can we get rid of any vector?

anonymous · Answer

Can we get rid of any vector?
@Zarkon This is basically what I'm asking :)

zarkon · Answer

yes

anonymous · Answer

That's weird. So you can have three different basis for the set of vectors?

anonymous · Answer

Does this mean that basis is not unique?

zarkon · Answer

correct the basis is not unique...there can be an infinite number of basis's

anonymous · Answer

Is there a condition where a basis will be unique?

zarkon · Answer

|dw:1336751060283:dw||dw:1336751083709:dw|

both a basis for $\mathbb{R}^2$

zarkon · Answer

you would have to be pretty restrictive

anonymous · Answer

What a bais for a vector space? Can that be infinite as well?

zarkon · Answer

yes

anonymous · Answer

and you can do this by multiplying the basis by a scalar?

anonymous · Answer

to get infinite basis?

zarkon · Answer

it is really too complicated to talk about here. I'd recommended you read http://en.wikipedia.org/wiki/Banach_space