document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=460760>Question 460760</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that if s=(v1,v2,v3) is a linearly dependent set of vectors in a vector space V, and v4 is any vector in V that is not in S, then {v1,v2,v3,v4} is also linearly dependent  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #316071 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=316071\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let S = {   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B1%5D ALIGN=MIDDLE  ALT=\"v%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B2%5D ALIGN=MIDDLE  ALT=\"v%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B3%5D ALIGN=MIDDLE  ALT=\"v%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >    } be a linearly dependent set of vectors in V.\r<BR>\n" );
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document.write( "This means that there are scalars <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B1%5D ALIGN=MIDDLE  ALT=\"c%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B2%5D ALIGN=MIDDLE  ALT=\"c%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B3%5D ALIGN=MIDDLE  ALT=\"c%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, not all of them zero, such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B1%5D%2Av%5B1%5D+%2B+c%5B2%5D%2Av%5B2%5D+%2B+c%5B3%5D%2Av%5B3%5D+=+theta ALIGN=MIDDLE  ALT=\"c%5B1%5D%2Av%5B1%5D+%2B+c%5B2%5D%2Av%5B2%5D+%2B+c%5B3%5D%2Av%5B3%5D+=+theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  But then, we would also have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B1%5D%2Av%5B1%5D+%2B+c%5B2%5D%2Av%5B2%5D+%2B+c%5B3%5D%2Av%5B3%5D+%2B+0%2Av%5B4%5D+=+theta ALIGN=MIDDLE  ALT=\"c%5B1%5D%2Av%5B1%5D+%2B+c%5B2%5D%2Av%5B2%5D+%2B+c%5B3%5D%2Av%5B3%5D+%2B+0%2Av%5B4%5D+=+theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >, letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B4%5D+=+0 ALIGN=MIDDLE  ALT=\"c%5B4%5D+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >; and not all of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B1%5D ALIGN=MIDDLE  ALT=\"c%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B2%5D ALIGN=MIDDLE  ALT=\"c%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B3%5D ALIGN=MIDDLE  ALT=\"c%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5B4%5D ALIGN=MIDDLE  ALT=\"c%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > are equal to zero.  Hence {   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B1%5D ALIGN=MIDDLE  ALT=\"v%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B2%5D ALIGN=MIDDLE  ALT=\"v%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B3%5D ALIGN=MIDDLE  ALT=\"v%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >   , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B4%5D ALIGN=MIDDLE  ALT=\"v%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > }  is also a linearly dependent set. </SPAN>\n" );
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