document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185219>Question 1185219</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Express the column matrix b as a linear combination of the columns of A. (Use A1, A2, and A3 respectively for the columns of A.)<BR>\n" );
document.write( "A = \r<BR>\n" );
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document.write( "3	1	2	\r<BR>\n" );
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document.write( "1	−1	−3<BR>\n" );
document.write( ",  \r<BR>\n" );
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document.write( " b =  −1	<BR>\n" );
document.write( "           3<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #816016 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=816016\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28matrix%282%2C1%2C3%2C1%29%29%2Bq%28matrix%282%2C1%2C1%2C-1%29%29%2Br%28matrix%282%2C1%2C2%2C-3%29%29=%28matrix%282%2C1%2C-1%2C3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This gives us two equations in the three variables p, q, and r, so we will get either no solution or an infinite family of solutions.<br><BR>\n" );
document.write( "[1] <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p%2Bq%2B2r=-1 ALIGN=MIDDLE  ALT=\"3p%2Bq%2B2r=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "[2] <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p-q-3r=3 ALIGN=MIDDLE  ALT=\"p-q-3r=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Adding the two equations eliminates q:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4p-r=2 ALIGN=MIDDLE  ALT=\"4p-r=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Solve that for r:<br><BR>\n" );
document.write( "[3] <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=4p-2 ALIGN=MIDDLE  ALT=\"r=4p-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "At this point, since the problem asks us to express B as a linear combination of the columns of A, we can simply choose a value for p, use the last equation to find the corresponding value of r, and then use either of our two original equations to find the value of q.<br><BR>\n" );
document.write( "Choose p=1<BR>\n" );
document.write( "Then r=4(1)-2=2<BR>\n" );
document.write( "And q=p-3r-3=1-6-3=-8<br><BR>\n" );
document.write( "So one solution is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%28matrix%282%2C1%2C3%2C1%29%29%2B-8%28matrix%282%2C1%2C1%2C-1%29%29%2B2%28matrix%282%2C1%2C2%2C-3%29%29=%28matrix%282%2C1%2C-1%2C3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "If we wanted to find parametric representation of the complete family of solutions, we could use [2] and [3] to express q in terms of p:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p-q-3r=3 ALIGN=MIDDLE  ALT=\"p-q-3r=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p-q-3%284p-2%29=3 ALIGN=MIDDLE  ALT=\"p-q-3%284p-2%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p-q-12p%2B6=3 ALIGN=MIDDLE  ALT=\"p-q-12p%2B6=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q=p-12p%2B6-3 ALIGN=MIDDLE  ALT=\"q=p-12p%2B6-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q=-11p%2B3 ALIGN=MIDDLE  ALT=\"q=-11p%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The family of solutions is then described by<br><BR>\n" );
document.write( "(p,q,r) = (p,-11p+3,4p-2)<br><BR>\n" );
document.write( "We can then verify the correctness of that parametric representation by plugging those expressions for p, q, and r in [1] and [2]:<br><BR>\n" );
document.write( "[1]: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p%2Bq%2B2r+=+3%28p%29%2B%28-11p%2B3%29%2B2%284p-2%29+=+3p-11p%2B3%2B8p-4+=+-1 ALIGN=MIDDLE  ALT=\"3p%2Bq%2B2r+=+3%28p%29%2B%28-11p%2B3%29%2B2%284p-2%29+=+3p-11p%2B3%2B8p-4+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "[2]: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p-q-3r+=+%28p%29-%28-11p%2B3%29-3%284p-2%29+=+p%2B11p-3-12p%2B6+=+3 ALIGN=MIDDLE  ALT=\"p-q-3r+=+%28p%29-%28-11p%2B3%29-3%284p-2%29+=+p%2B11p-3-12p%2B6+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );