document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=33284>Question 33284</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0387984550/algebrahomeworkh>Linear Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Let u={(x1,x2,x3)in R3|x1+x2+x3=a} for a in R3 fixed. Show the u is a vector subspace if and only if a=0.\r<BR>\n" );
document.write( "\n" );
document.write( "This abstract and I'm struggling on the abstract ideas in linear algebra.  <BR>\n" );
document.write( "Thank you for any help you can give me!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #19762 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=19762\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> BEFORE I ANSWER THIS LET ME ASK YOU ,YOU TOOK THIS<BR>\n" );
document.write( "ABSTRACT ALGEBRA BY YOUR CHOICE OR IT IS COMPULSORY IN YOUR COURSE OF<BR>\n" );
document.write( "STUDY.WHY I AM ASKING YOU THIS IS BECAUSE,YOU SAY YOU ARE STRUGGLING WITH THIS ABSTRACT COURSE AND INDEED THIS IS A DRY AND HIGHLY<BR>\n" );
document.write( "THEORETICAL SUBJECT AND UNLESS ONE HAS AN APTITUDE FOR THIS<BR>\n" );
document.write( "THEORETICAL STUDY , ONE IS LIKELY TO GET INTO PROBLEM IN ANSWERING<BR>\n" );
document.write( "THESE QUESTIONS.SO FAR THESE ARE LITTLE  ELEMENTARY PROBLEMS BUT IF THEY<BR>\n" );
document.write( "GET INTO ADVANCED TOPICS ,IT WILL BE MORE DIFFICULT.A COURSE IN<BR>\n" );
document.write( "APPLIED MATHS ON THE OTHERHAND WOULD BE EASIER IN GENERAL.FOR MANY NOT<BR>\n" );
document.write( "INCLINED FOR THEORETICAL ABSTRACTS.<BR>\n" );
document.write( "OK LET US GET ON <BR>\n" );
document.write( "THE NECESSARY AND SUFFICIENT CONDITION FOR U TO BE A SUBSPACE IS THAT IF P AND Q ARE ANY 2 ELEMENTS OF U,AND C AND D ARE ANY 2 ELEMENTS IN R3 THEN CP+DQ SHOULD BE AN ELEMENT OF U.<BR>\n" );
document.write( "SO LET P (X1,X2,X3)AND Q (Y1,Y2,Y3)BE 2 ELEMENTS OF U.HENCE WE HAVE FROM THE GIVEN CONDITION <BR>\n" );
document.write( "X1+X2+X3=Y1+Y2+Y3=A......................................I<BR>\n" );
document.write( "THEN....CP+DQ =C(X1,X2,X3)+D(Y1,Y2,Y3)=(CX1,CX2,CX3)+(DY1,DY2,DY3)<BR>\n" );
document.write( "=[(CX1+DY1),(CX2+DY2),(CX3+DY3)]<BR>\n" );
document.write( "NOW FOR THIS TO BE IN U ,IT SHOULD SATISFY THE GIVEN CONDITION .....NAMELY..<BR>\n" );
document.write( "CX1+DY1+CX2+DY2+CX3+DY3=A<BR>\n" );
document.write( "C(X1+X2+X3)+D(Y1+Y2+Y3)=A...FROM I....WE GET<BR>\n" );
document.write( "CA+DA=A<BR>\n" );
document.write( "A(C+D)=A...................NOW SINCE C&D ARE ANY 2 ELEMENTS IN R.....C+D CANNOT BE TAKEN AS EQUAL TO 1.HENCE THIS CAN BE SATISFIED IF AND ONLY IF A=0<BR>\n" );
document.write( "FOR IF A=0 THEN WE HAVE 0*(C+D)=0..HENCE U IS A SUBSPACE.<BR>\n" );
document.write( "IF A IS NOT EQUAL TO ZERO,WE SHOWED ABOVE THAT U IS NOT A SUB SPACE.\r<BR>\n" );
document.write( "\n" );
document.write( "HENCE U WILL BE SUBSPACE IF AND ONLY IF A=0 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );