Ordered Basis

1) Let S be an ordered basis for n-dimensional vector space V. Show that if {w1, w2, ..., wk} is a linearly independent set of vectors in V, then {[w1]s, [w2]s,...,[wk]s} is a linearly independent set of vectors in R^n.

What I got so far is

w1 = a1V1 + a2V2 + ... + anVn

so, [w1]s =

[a1

a2

...

an]

The same thing for w2, [w2]s and wk, [wk]s.

My question how to go from there?

2) Let S and T be two ordered bases of an n-dimensional vector space V. Prove that the transition matrix from T - coordinates to S - coordinates is unique. That is, if A,B belong to Mnn both satisfy A[v]T = [V]S and B[V]T = [v]S for all v belong to V, then A = B.

My approach for this question is that

Let S = {v1, v2, vn}

Let T = {w1, w2, wn}

Av = a1v1+a2v2+...+anvn

v = b1w1+b2w2+...+bnwn

Aa1v1 + Aa2v2+ ... +Aanvn

a1(Av1) + a2(Av2)+...+an(Avn)

= b1w1+b2w2+...+bn(wn)

Am I going the right direction? If no, how should I approach? If yes, how should I move from here?

Edited April 11, 201114 yr by hkus10