document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=32261>Question 32261</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given V=R^2 with \"non-standard\" operations: for u=(x1,y1) and v=(x2,y2)in R^2, and c(real number),<BR>\n" );
document.write( "u(+)v=[(x1^1/3 + x^2^1/3)^3, (y1^1/3 + y2^1/3)^3] and<BR>\n" );
document.write( "c(*)v=(c^3*x2, c^3*y2), where (*)and(+)anr non-standard operations,\r<BR>\n" );
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document.write( "I want to prove that V form a vector space, using 10 vector space axioms. But, I can't figure out how to handle the exponents (raised to the third power) in the operations...Also, I am confused about the \"non-standard\"...Can I still use some real number for x and y (like x=1, y=1)and 'c' when varifying the closure of the axioms? How exactly are \"standard\" and \"non-standard\" different when using the ten vector space axioms? For 'distributive property', could I use another vector w=(x3,y3)? <BR>\n" );
document.write( "Would you help me with this problem??? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #18796 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=18796\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I AM ANSWERING YOUR DOUBTS AND LEAVING THE SOLUTION TO YOU.IF YOU HAVE STILL DIFFICULTY PLEASE COME BACK AND WE SHALL SOVE IT FOR YOU...OK?<BR>\n" );
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document.write( "Given V=R^2 with \"non-standard\" operations: for u=(x1,y1) and v=(x2,y2)in R^2, and c(real number),<BR>\n" );
document.write( "u(+)v=[(x1^1/3 + x^2^1/3)^3, (y1^1/3 + y2^1/3)^3] and<BR>\n" );
document.write( "c(*)v=(c^3*x2, c^3*y2), where (*)and(+)anr non-standard operations,<BR>\n" );
document.write( "I want to prove that V form a vector space, using 10 vector space axioms.<BR>\n" );
document.write( "GOOD.....PROCEED..THOUGH I AM NOT SURE ABOUT YOUR COUNT OF 10 AXIOMS..ANY WAY I HOPE THEY INCLUDE ALL REQUIRED STIPULATIONS.<BR>\n" );
document.write( " But, I can't figure out how to handle the exponents (raised to the third power) in the operations...<BR>\n" );
document.write( "SAME WAY AS YOU DO IN NORMAL ALGEBRA..THERE IS NO DIFFERENCE.THOUGH U AND V ARE VECTORS WITH SPECIAL PROPERTIES AS DEFINED,INDIVIDUALLY....C,X1,Y1 ETC...YOU CAN TREAT AS IN NORMAL MANNER.<BR>\n" );
document.write( "Also, I am confused about the \"non-standard\"...<BR>\n" );
document.write( "THIS WORD IS USED TO TELL THAT (+) IS NOT NORMAL + OR ADDITION...AS WE UNDERSTAND AND SIMILARLY (*) IS NOT NORMAL * OR MULTIPLICATION AS WE UNDERSTAND.YOU HAVE TO USE THE GIVEN FORMULAE TO DO THESE OPERATIONS OF (+) AND (*).BUT THE FORMULAE GIVEN FOR THOSE SPECIAL OR NON STANDARD OPERATION CONNECTING C,X1,Y1 ETC..HAS ^,+AND * WHICH ARE NORMAL OPERATIONS OF EXPONENTIATION,ADDITION AND MULTIPLICATION W.R.T THOSE VARIABLES.<BR>\n" );
document.write( "Can I still use some real number for x and y (like x=1, y=1)and 'c' when varifying the closure of the axioms?<BR>\n" );
document.write( "SURE ..THAT IS WHAT I HAD WRITTEN ABOVE.<BR>\n" );
document.write( " How exactly are \"standard\" and \"non-standard\" different when using the ten vector space axioms?<BR>\n" );
document.write( "I HOPE I EXPLAINED THIS IN DETAIL ABOVE..IF YOU ARE DEALING WITH U,V,OR W...THE VECTORS ...YOU HAVE TO USE THE GIVEN FORMULAE FOR * OR + AS SHOWN DISTINCTLY BY (*),(+)......BUT WHEN YOU ARE DEALING WITH C,X1,Y1,ETC...INDIVIDUALLY,YOU CAN DO NORMAL ADDITION MULTIPLICATION ETC...<BR>\n" );
document.write( " For 'distributive property', could I use another vector w=(x3,y3)?<BR>\n" );
document.write( "SURE..YOU SIMPLY WRITE LET W (X3,Y3) BE A VECTOR AND PROCEED...<BR>\n" );
document.write( "Would you help me with this problem???<BR>\n" );
document.write( "I TRUST I DID THAT TO YOUR SATISFACTION...GOOD LUCK!IF YOU NEED MORE PLEASE COME BACK </SPAN>\n" );
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