document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=31686>Question 31686</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0471669598/algebrahomeworkh>Elementary Linear Algebra with Applications</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I was given a homework sheet in class and have no idea what chapter it is from.  Let A be an nxn matrix and suppose that v1 and v2 are nonzero vectors in R^n that satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Av1=v1 ALIGN=MIDDLE  ALT=\"Av1=v1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Av2=2v2 ALIGN=MIDDLE  ALT=\"Av2=2v2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Prove that the set {v1, v2} is linearly independent.\r<BR>\n" );
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document.write( "Thank you for your help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #18465 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=18465\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> LET US PROVE THIS BY REDUXIO-AD-ABSURDUM OR BY CONTRADICTION.<BR>\n" );
document.write( "LET US ASSUME THAT V1 AND V2 ARE NOT LINEARLY INDEPENDENT.<BR>\n" );
document.write( "THAT IS THEY ARE LINEARLY DEPENDENT.<BR>\n" );
document.write( "SO LET V1=KV2 WHERE K IS A SCALAR.<BR>\n" );
document.write( "WE ARE GIVEN <BR>\n" );
document.write( "AV1=V1 ........AKV2=V1....OR......KAV2=V1<BR>\n" );
document.write( "BUT AV2=2V2<BR>\n" );
document.write( "SO K*2V2=V1<BR>\n" );
document.write( "2KV2=V1=2KV2...BUT OUR ASSUMPTION IS V1=KV2<BR>\n" );
document.write( "HENCE KV2=2KV2...OR....K=2K...OR....K=0<BR>\n" );
document.write( "SO...V1=0V2=0...BUT V1 IS NONZERO VECTOR AS PER HYPOTHESIS.<BR>\n" );
document.write( "HENCE THIS IS NOT CORRECT.SO OUR ASSUMPTION THAT V1 AND V2 ARE LINEARLY DEPENDENT IS WRONG.HENCE V1 AND V2 ARE LINEARLY INDEPENDENT.  </SPAN>\n" );
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