if S={u1,u2,.....,un} and S'={v1,v2,....,vn} be two bases far a vector space V then v1=au1+bu2+......+zun ie every element of V(it can be v1 which is also the element of 2nd basis) can be written as a linear combination of basis vectors u1,u2,....un. how is it possible that a linearly independent vector, i.e v1 in this case, can b written as a linear combination of other linearly independent vectors. how???

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- cristiann

The concept of linear independence doesn't refer to a vector - it refers to a family of one or more vectors:
So, in your statement, you have:
S is linear independent (because it is a base)
also S' is linear independent (because it is a base)
from the relation
v1=au1+bu2+......+zun
you get that the family/set {v1,u1,u2,.....,un} is not linear independent (assuming at least one scalar not null), which is not in conflict with the previous statements

- anonymous

here aqain a que arises that {v1,u1,u2,.....,un} is not linear independent is sensible according to theorem ''if S={u1,u2,.....,un} is a basis for V then any collection of m vectors where m>n is linearly dependent.'' but it becomes senseless when we see it another way that v1 is linearly independent bcz it is the element of basis that's why its coefficient must be zero. and u1,u2....,un are also linearly independent thats why these coefficients are also zero then how can you say that {v1,u1,u2,.....,un} is not linear independent (assuming at least one scalar not null) where all the scalars are already zero. how???

- cristiann

OK ... I'll try to rephrase ...
A set is called "dependent" if an element of the set is a linear combination of the other elements of the set.
A set which is not dependent is called "independent"
Note that we are talking about sets of vectors, not about vectors.
You cannot say about a vector v that is dependent/independent ... it's meaningless.
But you can say about the set {v} that it is dependent/independent (iff v not=0)
When v1 is in S', the set {v1} is independent (as a subset of an independent set)
Now if you take the base S,
if v1 belongs to S, then {v1,u1,u2,.....,un}=S (and so S remains independent)
if v1 is not in S, then {v1,u1,u2,.....,un} is dependent, because S is a basis so v1 is a linear combination of the other vectors of the family.

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