document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=8896>Question 8896</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1; let  v  be the  space of  all  polynomials over a field F of  degree less  or equal to 3 <BR>\n" );
document.write( "          let  D be  a  diff. operator  from  v to v <BR>\n" );
document.write( "    1; find  the  matrix A of D with  respect  to  the  basis {1,,x,x.x,x.x.x} of v . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #4855 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=4855\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> With your level,it seems that you should know how to type<BR>\n" );
document.write( " {1,x,x^2,x^3} instead of the ugly form you have given.<BR>\n" );
document.write( " <BR>\n" );
document.write( " Let D: V-->V (capital in general. It is important to follow some convention<BR>\n" );
document.write( "               in math.)\r<BR>\n" );
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document.write( " Any linear transformation is unquely determined by its values on the <BR>\n" );
document.write( " basis.\r<BR>\n" );
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document.write( " Now, D(1) =0, (corresponding to the 1st column of A)<BR>\n" );
document.write( "  D(x) = 1 , (the 2nd column of A)<BR>\n" );
document.write( "  D(x^2) = 2x = 2 * x, (the 3rd column of A) and<BR>\n" );
document.write( "  D(x^3) = 3x^2 = 3* x^2. (the 4th column of A)<BR>\n" );
document.write( "   <BR>\n" );
document.write( "Hence, the matrix A od D wrtto the basis {1,x,x^2,x^3}<BR>\n" );
document.write( " is <BR>\n" );
document.write( "    [0 1 0 0<BR>\n" );
document.write( "     0 0 2 0<BR>\n" );
document.write( "     0 0 0 3<BR>\n" );
document.write( "     0 0 0 0 ]\r<BR>\n" );
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document.write( " Note: D(xi) = E Aik xk (E means summation k from 1 to 3<BR>\n" );
document.write( "                        {xi} is a basis)\r<BR>\n" );
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document.write( " If you have trouble understanding,try to review the def. about<BR>\n" );
document.write( " the matrix representaion for a linear transformation carefully.\r<BR>\n" );
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document.write( " After all, this is a very basic problem in linear algebra.\r<BR>\n" );
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document.write( " Kenny<BR>\n" );
document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );