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)
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