# Coordinate vector basis proving question

1. Mar 18, 2010

### gtse

1. The problem statement, all variables and given/known data
Let S = {v1,v2,...,vn} be a basis for an n-dimensional vector space V. Show that {[v1]s,[v2]s,...[vn]s} is a basis for Rn.
Here [v]s means the coordinate vector with respect to the basis S.

2. Relevant equations
[v]s is the coordinate vector with respect to the basis S.

3. The attempt at a solution
S={v1..vn} is a basis and must be linearly independent.
Any vector v in S then is a unique linear combination of the vectors in S, so v=a1v1+a2v2+...+anvn.
Since [v]s in general = (a1,a2,...an), then every [vi]s where i = 1 .. n has a unique (a1,a2,...,an) and so the basis {[v1]s,...,[v2]s} will be linearly independent and thus form a basis for Rn.

I have no answers to verify with, so I would like to know if I have answered it correctly. I am extremely weak with anything to do with proving so any assistance would be greatly appreciated, :).