document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=18994>Question 18994</A>: <I><SPAN STYLE=\"word-break: break-all;\"> is the following a linear transformation from R^3 to R^2?<BR>\n" );
document.write( "L(x1,x2,x3) = (x2-x3,x1+2)  ?\r<BR>\n" );
document.write( "\n" );
document.write( "so i know how to prove that this is not closed under scalar multiplication and addition (due to the \"2\" in the second entry).  so then is my answer just \"no, it is not a linear transformation from R^3 to R^2 because it is not closed under scalar multiplication.\"  ?\r<BR>\n" );
document.write( "\n" );
document.write( "what about linear transformation from R^3 to R^4: L(x1,x2,x3) = (x2-x3,x1*x2,2x2-x1,x3+x2)  ?\r<BR>\n" );
document.write( "\n" );
document.write( "for this one, i dont quite understand how to go about it because the second entry is a product of two numbers and not just a sum.  so then whould it count as being closed under scalar multiplication and addition?  plus i remember seeing somewhere if R^m ---> R^n, then m must be greater than or equal to n.  so then by this rule, the answer is automatically no because we are going from R^3 to R^4 and (obviously) 4 is greater than 3.  i'm just not sure about all of this...\r<BR>\n" );
document.write( "\n" );
document.write( "thank you so much for your help : ) i greatly appreciate it!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #9136 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=9136\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  a linear transformation from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E3 ALIGN=MIDDLE  ALT=\"R%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2+ ALIGN=MIDDLE  ALT=\"R%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >?<BR>\n" );
document.write( " L(x1,x2,x3) = (x2-x3,x1+2) ? <BR>\n" );
document.write( " (i) It is still closed under scalar multiplication and addition  but<BR>\n" );
document.write( " not linear. Simly, because L(0,0,0) != (0,0).\r<BR>\n" );
document.write( "\n" );
document.write( " Since, for any   a linear transformation L, L(0) = L(0+0)= L(0) + L(0)<BR>\n" );
document.write( " so L(0) = 0 for zero vector. \r<BR>\n" );
document.write( "\n" );
document.write( " In general, such L <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5Em ALIGN=MIDDLE  ALT=\"R%5Em\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5En+ ALIGN=MIDDLE  ALT=\"R%5En+\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if each n component<BR>\n" );
document.write( " of L is a linear function. Now the 2nd component x1+2 is not linear.<BR>\n" );
document.write( " [You idea was Ok with wrong reason.]\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so i know how to prove that this is not closed under scalar multiplication and addition (due to the \"2\" in the second entry). so then is my answer just \"no, it is not a linear transformation from R^3 to R^2 because it is not closed   <BR>\n" );
document.write( "under scalar multiplication.\" ? \r<BR>\n" );
document.write( "\n" );
document.write( " NO !!!!\r<BR>\n" );
document.write( "\n" );
document.write( "what about linear transformation from R^3 to R^4: L(x1,x2,x3) = (x2-x3,x1*x2,2x2-x1,x3+x2) ? \r<BR>\n" );
document.write( "\n" );
document.write( " No, since the 2nd component x1*x2 is not linear.<BR>\n" );
document.write( " More precisely,<BR>\n" );
document.write( " L(x1,x2,x3) = (x2-x3,x1*x2,2x2-x1,x3+x2) <BR>\n" );
document.write( " <BR>\n" );
document.write( " Since (1,1,0) = (1,0,0) +(0,1,0).<BR>\n" );
document.write( " L(1,1,0) = (1, 1,1, 1)<BR>\n" );
document.write( " L(1,0,0) = (0, 0, -1, 0) and L(0,1,0) = (1, 0, 2, 1) \r<BR>\n" );
document.write( "\n" );
document.write( " but L(1,0,0) + L(0,1,0) = (1,0,1,1) != L(1,1,0)\r<BR>\n" );
document.write( "\n" );
document.write( " Hence,L is not a linear transformation.\r<BR>\n" );
document.write( "\n" );
document.write( " Another wrong idea:<BR>\n" );
document.write( "\" if<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+R%5Em ALIGN=MIDDLE  ALT=\"+R%5Em\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5En ALIGN=MIDDLE  ALT=\"R%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then m must be greater than or equal to n. \"\r<BR>\n" );
document.write( "\n" );
document.write( " No, m ,n can be any two non-negative integers, not restricting to m<=n.\r<BR>\n" );
document.write( "\n" );
document.write( " Good luck !\r<BR>\n" );
document.write( "\n" );
document.write( " Kenny<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );