document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=31476>Question 31476</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let S={v1, v2, v3}where v1^T=[1 0]   v2^T=[0 1]  v3^T=[-1 1]<BR>\n" );
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document.write( "Find every subset of S that is a basis for R^2.\r<BR>\n" );
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document.write( "I have the matrix<BR>\n" );
document.write( "[1 0 -1 0]<BR>\n" );
document.write( "[0 1 1 0]<BR>\n" );
document.write( "where x1-x3=0 and x2+x3=o and one basis of {v1, v2} but I don't know where to go from there. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #18146 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=18146\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let S={v1, v2, v3}where v1^T=[1 0]   v2^T=[0 1]  v3^T=[-1 1]<BR>\n" );
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document.write( "Find every subset of S that is a basis for R^2.<BR>\n" );
document.write( "BASIS DEMANDS 2 THINGS...<BR>\n" );
document.write( "1. THEY SHOULD BE LINMEARLY INDEPENDENT<BR>\n" );
document.write( "2.THEY SHOULD SPAN THE ENTIRE SPACE.<BR>\n" );
document.write( "FOR FOR 2 DIMENSIONS (R^2)THERE WILL BE  2 AND ONLY 2 INDEPENDENT VECTORS WHICH SATISFY BOTH THE ABOVE REQUIREMENTS.HENCE WE NEED TO FIND ONLY SUBSETS OF 2 ELEMENTS...THEY COULD BE <BR>\n" );
document.write( "1...V1V2....OR.....2..V1V3....OR.....3...V2V3<BR>\n" );
document.write( "FOR TESTING INDEPENTNESS AND SPANNING THE ENTIRE SPACE HERE WE ONLY NEED TO CHECK INDEPENTNESS AS THEN IN R^2 , 2 INDEPENDENT VECTORS WILL CERAINLY SPAN THE ENTIRE SPACE.<BR>\n" );
document.write( "1.V1V2,,,<BR>\n" );
document.write( "LET AV1+BV2=0<BR>\n" );
document.write( "A(1,0)+B(0,1)=0<BR>\n" );
document.write( "THIS CAN HAPPEN ONLY WHEN A=B=0..SO THEY ARE INDEPENDENT...HENCE THIS IS A BASIS.<BR>\n" );
document.write( "2.V1V3<BR>\n" );
document.write( "A(1,0)+B(-1,1)=0...THIS ALSO GIVES A=B=0...SO THEY ARE INDEPENDENT...HENCE THIS IS A BASIS.<BR>\n" );
document.write( "3.V2V3<BR>\n" );
document.write( "A(0,1)+B(-1,1)=0.....ALSO GIVES A=B=0...SO THEY ARE INDEPENDENT...HENCE THIS IS A BASIS.<BR>\n" );
document.write( "SO ALL 3 SUBSETS CAN FORM A BASIS IN R^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I have the matrix<BR>\n" );
document.write( "[1 0 -1 0]<BR>\n" );
document.write( "[0 1 1 0]<BR>\n" );
document.write( "where x1-x3=0 and x2+x3=o and one basis of {v1, v2} but I don't know where to go from there. </SPAN>\n" );
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