document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=698554>Question 698554</A>: <I><SPAN STYLE=\"word-break: break-all;\"> show whether the following vectors are dependent or independent                       u1(1.1.2.2)  u2(1.2.3.2)   u3(-1.1.2.1)  u4(2.2.2.1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #430989 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20055)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=430989\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "u1(1,1,2,2)  u2(1,2,3,2)   u3(-1,1,2,1)  u4(2,2,2,1)\r<BR>\n" );
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document.write( "We form the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%284%2C1%2Cu%5B1%5D%2Cu%5B2%5D%2Cu%5B3%5D%2Cu%5B4%5D%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%284%2C1%2Cu%5B1%5D%2Cu%5B2%5D%2Cu%5B3%5D%2Cu%5B4%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%284%2C4%2C%0D%0A%0D%0A1%2C1%2C2%2C2%2C%0D%0A1%2C2%2C3%2C2%2C%0D%0A-1%2C1%2C2%2C1%2C%0D%0A2%2C2%2C2%2C1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We reduce the matrix to row-echelon form\r<BR>\n" );
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document.write( "R2-R1->R2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%284%2C4%2C1%2C1%2C2%2C2%2C%0D%0A0%2C1%2C1%2C0%2C%0D%0A-1%2C1%2C2%2C1%2C%0D%0A2%2C2%2C2%2C1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "R1+R3->R3\r<BR>\n" );
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document.write( "-2·R1+T4->R4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%284%2C4%2C%0D%0A1%2C1%2C2%2C2%2C%0D%0A0%2C1%2C1%2C0%2C%0D%0A0%2C2%2C4%2C3%2C%0D%0A0%2C0%2C-2%2C-3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-2R2+R3->R3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%284%2C4%2C%0D%0A1%2C1%2C2%2C2%2C%0D%0A0%2C1%2C1%2C0%2C%0D%0A0%2C0%2C2%2C3%2C%0D%0A0%2C0%2C-2%2C-3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "R3+R4->R4\r<BR>\n" );
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document.write( "The all-zero row at the bottom tells us the<BR>\n" );
document.write( "matrix has rank 3 and nullity 1, thus those 4 <BR>\n" );
document.write( "vectors are dependent.\r<BR>\n" );
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document.write( "Edwin<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"> </SPAN>\n" );
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