Let V be a finite n-dim vector space
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
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)
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