Thread: Linear Algebra- Basis

Let V be a finite n-dim vector space

v1,v2,....,vn
w1,w2,...,wn

be two set of basis

Let v be a vector in V be represented in this basis as follows:
v = a1.v1 + a2.v2 + .... + an.vn
also
v = a1.w1 + a2.w2 + ... + an.wn
(note the coefficients a1,a2,...,an are same for both basis)

Can we conclude that v1=w1; v2 = w2' ....vn = wn from this?

Is yes - why?
(obviously I assume v is not equal to 0)

Originally Posted by aman_cc

Let V be a finite n-dim vector space

v1,v2,....,vn
w1,w2,...,wn

be two set of basis

Let v be a vector in V be represented in this basis as follows:
v = a1.v1 + a2.v2 + .... + an.vn
also
v = a1.w1 + a2.w2 + ... + an.wn
(note the coefficients a1,a2,...,an are same for both basis)

Can we conclude that v1=w1; v2 = w2' ....vn = wn from this?

Is yes - why?
(obviously I assume v is not equal to 0)

no! for example in let be the standard basis. then is also a basis for and

Thanks - a very neat counter example. But does my equation put some kind of constraint on the basis - I mean is there a special relation between v1,v2...vn and w1,w2,...,wn for such a thing to happen? I am kind of confused here

Originally Posted by aman_cc

Thanks - a very neat counter example. But does my equation put some kind of constraint on the basis - I mean is there a special relation between v1,v2...vn and w1,w2,...,wn for such a thing to happen? I am kind of confused here

what i can say now is that we'll have this situation whenever we have an matrix with (the base field) with these properties that and

to see that let be any basis for and define since the set is a basis for also since there

exists a vector such that which is equivalent to