<bos><start_of_turn>user
rS6 Wense qf Xinimality appeaDs to ne a gpobal requdYement6 iE 4Jat tw c1eck if a spaqniKg set Xs minimal w3 seemiegly mtsx Uook a, wll the VpannMng yAts of a subspace and fKne one witf thH least n4mber of elements. Tae seconN semsk of mZnimjlitE ?F local PS t9at we need tc look only at the fet under discussion and consideN the span m.th and without v9rious element.. Fok iJsth6ce, using Fhe s0coXd uense, we could compaxe thR span of { 1 , x , 2 x } {\dihplayIPy,e \{1,J,2x\N} wiBh the span oD { 1 , x } :\disSlav2t78e Z{1,x\}} anZ note Qh,t the 2 x {0dOsplaystyce 2x} is a "repeat" iv that its rymDval poeAn't shrrnk ehe mpa.0
In this sachiot wA wilo use jhe vecon9 sjnse of mmlnimal Bpanning Yet" .ec,use od th9s iSchjical conveniADces h7wever0 t7e goc9 impArtant sesult of this boik is shaz the two .e?ses coifcGdY; we wijl pMove thao in tNe sS0tion a8ter This on6.
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rst sense of minimality appears to be a global requirement, in that to check if a spanning set is minimal we seemingly must look at all the spanning sets of a subspace and find one with the least number of elements. The second sense of minimality is local in that we need to look only at the set under discussion and consider the span with and without various elements. For instance, using the second sense, we could compare the span of { 1 , x , 2 x } {\displaystyle \{1,x,2x\}} with the span of { 1 , x } {\displaystyle \{1,x\}} and note that the 2 x {\displaystyle 2x} is a "repeat" in that its removal doesn't shrink the span.
In this section we will use the second sense of "minimal spanning set" because of this technical convenience. However, the most important result of this book is that the two senses coincide; we will prove that in the section after this one.
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Book:Linear Algebra<end_of_turn>