document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=30580>Question 30580</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I want to find about(three things)the set, W={(1,0,0),(0,1,0),(0,0,1),(-7,8,-9)}<BR>\n" );
document.write( "Is this linearly dependent? Does this spans R3? Is this a basis for R3?<BR>\n" );
document.write( "I think this set spansR3 because the first three vectors span R3, so adding the last one does not change it. Also, I think it is linearly dependent as there are nontrivial solutions. But, I am not sure if it is not a vasis for R3. Could you give me advice??? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #17264 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=17264\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I want to find about(three things)the set, W={A=(1,0,0),B=(0,1,0),C=(0,0,1),D=(-7,8,-9)}<BR>\n" );
document.write( "Is this linearly dependent?<BR>\n" );
document.write( "YES FOR REASON GIVEN BY YOU ..WE CAN SHOW AS FOLLOWS EASILY<BR>\n" );
document.write( "WE NOTICE THAT <BR>\n" );
document.write( "D=-7A+8B-9C ...OR<BR>\n" );
document.write( "WE ARE ABLE TO FIND <BR>\n" );
document.write( "PA+QB+RC+SD=0...(-7A+8B-9C-D=0),WHILE<BR>\n" );
document.write( "P,Q,R,S ARE NOT ALL EQUAL TO ZERO.HENCE THEY ARE LINEARLY DEPENDENT.<BR>\n" );
document.write( " Does this spans R3?<BR>\n" );
document.write( "YES<BR>\n" );
document.write( " Is this a basis for R3?<BR>\n" );
document.write( "FOR THIS YOU HAVE TO REFER DEFINITION OF BASIS..IT STIPULATES..W A SUBSET OF VECTOR SPACE R3 IS A BASIS FOR R3 IF<BR>\n" );
document.write( "1.W CONSISTS OF  LINEARLY INDEPENDENT VECTORS<BR>\n" );
document.write( "2.W GENERATES R3 OR EACH VECTOR IN R3 IS A LINEAR COMBINATION OF FINITE NUMBER OF ELEMENTS OF W .<BR>\n" );
document.write( "SO WE HAVE CONDITION 2.SATISFIED ..BUT NOT CONDITION 1..SO W IS NOT A BASIS. <BR>\n" );
document.write( "I think this set spansR3 because the first three vectors span R3,<BR>\n" );
document.write( "VERY GOOD<BR>\n" );
document.write( " so adding the last one does not change it.<BR>\n" );
document.write( "CORRECT<BR>\n" );
document.write( " Also, I think it is linearly dependent as there are nontrivial solutions. <BR>\n" );
document.write( "PERFECT AS WE SAW ABOVE <BR>\n" );
document.write( "But, I am not sure if it is not a vasis for R3. Could you give me advice???<BR>\n" );
document.write( "YOU HAVE DONE EXTREMELY WELL...KEEP IT UP..YOU JUST NEED A LITTLE POLISHING...GOOD LUCK. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );