document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=12549>Question 12549</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I'm having some trouble with the concept of linear operators. I think of them as functions on sets, but still can't imagine actual examples. Given the properties of linear operators, these should be simple ones (though I still haven't come up with answers), so I'm hoping you can give me some ideas to help me grasp the linear operator a bit better. Here we go! 1) I'm looking for a linear operator from V->V such that T^2=0 but T does not =0. Then, 2) Given two linear operators (say, T, U) from V-> V, I'm looking for TU=0 but UT does not =0. The generality of T and U just really throws me off for some reason. Thanks again for your time. I really 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 #6417 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=6417\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">   Linear operators are not merely functions on sets,they are linear transfomrations(preserving addition and scalar multiplication of vectors on vector space. \r<BR>\n" );
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document.write( " I feel very strange that you haven't mentioned vector space, or matrices.<BR>\n" );
document.write( " Without them, how can you start solving the questions ?\r<BR>\n" );
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document.write( " 1) I'm looking for a linear operator from V->V such that T^2=0 but T does not =0. <BR>\n" );
document.write( "  Sol: Define <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+T%3AR%5E2 ALIGN=MIDDLE  ALT=\"+T%3AR%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <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;\" > generated by<BR>\n" );
document.write( "   T(1,0) = (0,1) and T(0,1) = (0,0)<BR>\n" );
document.write( " [Note : (1,0) & (0,1) are standard basis of unitcolumn vectors]<BR>\n" );
document.write( "  Let B = {(1,0),(0,1)} <BR>\n" );
document.write( " The matrix A = [T}B of T associated with the basis B as<BR>\n" );
document.write( " [0 0] <BR>\n" );
document.write( " [1 0] <BR>\n" );
document.write( " or equivalently T(X) = AX for all column vector X in <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( " Clearly,you can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2+=+0+ ALIGN=MIDDLE  ALT=\"A%5E2+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > but A is not 0.<BR>\n" );
document.write( "  More precisely, T(a, b) = aT(1,0) + bT(0,1) = a(0,1) = (0,a)<BR>\n" );
document.write( " for all (a,b) in <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( " Check: Clearly,T is not the zero operator (why?) and<BR>\n" );
document.write( " we see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+T%5E2 ALIGN=MIDDLE  ALT=\"+T%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >(a,b) = T(0,a) = 0*T(1,0) = (0,0) for all real a,b<BR>\n" );
document.write( " 2) Given two linear operators (say, T, U) from V-> V, I'm looking for TU=0 but UT does not =0<BR>\n" );
document.write( " Sol: Similarly to the example in 1)<BR>\n" );
document.write( "      Let T ,U be two linear operators in <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;\" > defined by<BR>\n" );
document.write( "     T = (as matrix A) <BR>\n" );
document.write( "    [0 1]<BR>\n" );
document.write( "    [0 0] and<BR>\n" );
document.write( "    U = (as matrix B)<BR>\n" );
document.write( "    [1 0]<BR>\n" );
document.write( "    [0 0] then we have<BR>\n" );
document.write( "    TU =<BR>\n" );
document.write( "    [0 0]<BR>\n" );
document.write( "    [0 0] but UT =<BR>\n" );
document.write( "    [0 1]<BR>\n" );
document.write( "    [0 0] (not zero operator)<BR>\n" );
document.write( "  More precisely, define <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+T%3AR%5E2+ ALIGN=MIDDLE  ALT=\"+T%3AR%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <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;\" > by T(X) = AX <BR>\n" );
document.write( "  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+T%3AR%5E2 ALIGN=MIDDLE  ALT=\"+T%3AR%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <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;\" > by T(X) = BX where X is any column vector of <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( "  AX & BX are products of matrices.\r<BR>\n" );
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document.write( " Make sure you do understand the examples above and work hard. \r<BR>\n" );
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document.write( " Kenny<BR>\n" );
document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
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