hkus10 Posted April 11, 2011 Share Posted April 11, 2011 (edited) 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, 2011 by hkus10 Link to comment Share on other sites More sharing options...
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