document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=34258>Question 34258</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Problem: Is it possible for the columns of a 4x3 matrix to be linearly dependent? if so, give an example and demonstrate the dependence. If not, prove it.\r<BR>\n" );
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document.write( "I understand what linearly dependent means, but I don't know what kind of a matrix we should use to prove it without coming up with a clear example with numbers. Please help.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #20579 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=20579\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Problem: Is it possible for the columns of a 4x3 matrix to be linearly dependent?<BR>\n" );
document.write( "YES ,WE CAN ALWAYS WRITE TO SUIT THE REQUIREMENT..FOR..EXAMPLE...LET THE MATRIX A BE EQUAL TO <BR>\n" );
document.write( "1,2,3<BR>\n" );
document.write( "2,4,6<BR>\n" );
document.write( "3,6,9<BR>\n" );
document.write( "4,8,12<BR>\n" );
document.write( "THE COLUMNS ARE <BR>\n" );
document.write( "C1...........1,2,3,4<BR>\n" );
document.write( "C2...........2,4,6,8<BR>\n" );
document.write( "C3...........3,6,9,12<BR>\n" );
document.write( "WE FIND EACH ELEMENT OF C2=2*EACH CORRESPONDING ELEMENT OF C1<BR>\n" );
document.write( "WE FIND EACH ELEMENT OF C3=3*EACH CORRESPONDING ELEMENT OF C1<BR>\n" );
document.write( "SO THE COLUMNS ARE LINEARLY DEPENDENT....<BR>\n" );
document.write( "ANOTHER WAY TO MAKE SUCH MATRIX IS SAY A IS.............<BR>\n" );
document.write( "1,2,3<BR>\n" );
document.write( "4,5,9<BR>\n" );
document.write( "6,7,13<BR>\n" );
document.write( "8,9,17<BR>\n" );
document.write( "THE COLUMNS ARE<BR>\n" );
document.write( "C1...1,4,6,8<BR>\n" );
document.write( "C2...2,5,7,9<BR>\n" );
document.write( "C3...3,9,13,17<BR>\n" );
document.write( "WE FIND EACH ELEMENT OF C3=SUM OF EACH CORRESPONDING ELEMENT OF C1 AND C2<BR>\n" );
document.write( " if so, give an example and demonstrate the dependence. If not, prove it.<BR>\n" );
document.write( "I understand what linearly dependent means, but I don't know what kind of a matrix we should use to prove it without coming up with a clear example with numbers. Please help.<BR>\n" );
document.write( "IN GENERAL JUST MAKE NUMBERS IN ONE COLUMN TO BE SUM OR DIFFERENCE OF ELEMENTS IN OTHER 2 COLUMNS...SAY..A..IS...<BR>\n" );
document.write( "X,Y,X-Y<BR>\n" );
document.write( "P,Q,P-Q<BR>\n" );
document.write( "T,U,T-U<BR>\n" );
document.write( "E,F,E-F...ETC... </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );