document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=363408>Question 363408</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The space of polynomials in one variable, x, of degree at most 4, is mapped into itself by the linear map, L, that takes a typical polynomial P to the polynomial x(dP/dx)-4P, where dP/dx denotes the standard derivative of P.\r<BR>\n" );
document.write( "\n" );
document.write( "Thus: L(P)= x(dP/dx) - 4P\r<BR>\n" );
document.write( "\n" );
document.write( "1. Prove (show) that L is a linear map.\r<BR>\n" );
document.write( "\n" );
document.write( "2. Wat is the kernal of this map? (i.e. describe the polynomials that it contains)\r<BR>\n" );
document.write( "\n" );
document.write( "3. What is the dimension of the kernal?\r<BR>\n" );
document.write( "\n" );
document.write( "4. What is the dimension of the range of this map?\r<BR>\n" );
document.write( "\n" );
document.write( "5. Is there a polynomial of degree 2 that is not in the range of the map? give reasons for your answers to questions 2-5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #259187 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=259187\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28p%2Bq%29+=+xd%28p%2Bq%29%2Fdx-4%28p%2Bq%29+=+x%28dp%2Fdx+%2B+dq%2Fdx%29+-+4p-4q ALIGN=MIDDLE  ALT=\"L%28p%2Bq%29+=+xd%28p%2Bq%29%2Fdx-4%28p%2Bq%29+=+x%28dp%2Fdx+%2B+dq%2Fdx%29+-+4p-4q\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Hence<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28p%2Bq%29+=+xdp%2Fdx+%2B+xdq%2Fdx+-+4p-4q+=+xdp%2Fdx+-4p+%2B+xdq%2Fdx+-4q+=+L%28p%29+%2B+L%28q%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28alpha%2Ap%29+=+xd%28alpha%2Ap%29%2Fdx-4%28alpha%2Ap%29+=+alpha%2Axdp%2Fdx-alpha%2A4p+=+alpha%2A%28xdp%2Fdx+-+4p%29+=+alpha%2AL%28p%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a scalar.  Therefore L is a linear map.\r<BR>\n" );
document.write( "\n" );
document.write( "2.  Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28p%29+=+O ALIGN=MIDDLE  ALT=\"L%28p%29+=+O\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the zero polynomial.  This is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28p%29+=+xdp%2Fdx-4p+=+O ALIGN=MIDDLE  ALT=\"L%28p%29+=+xdp%2Fdx-4p+=+O\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Solving,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xdp%2Fdx+=+4p ALIGN=MIDDLE  ALT=\"xdp%2Fdx+=+4p\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dp%2Fp+=+4dx%2Fx ALIGN=MIDDLE  ALT=\"dp%2Fp+=+4dx%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lnp+=+4lnx+%2B+lnk ALIGN=MIDDLE  ALT=\"lnp+=+4lnx+%2B+lnk\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where k is a constant.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lnp+=+ln%28k%2Ax%5E4%29 ALIGN=MIDDLE  ALT=\"lnp+=+ln%28k%2Ax%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+k%2Ax%5E4 ALIGN=MIDDLE  ALT=\"p+=+k%2Ax%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Thus all polynomials of this form comprise kerL.\r<BR>\n" );
document.write( "\n" );
document.write( "3.  Consequently from part 2 above, dim ker L = 1.\r<BR>\n" );
document.write( "\n" );
document.write( "4.  Since dim kerL+ dim rangeL = 5 in this case, dim range L = 4.  This also becomes obvious if we put in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+a+%2Bbx%2B+cx%5E2+%2B+dx%5E3+%2B+ex%5E4 ALIGN=MIDDLE  ALT=\"p+=+a+%2Bbx%2B+cx%5E2+%2B+dx%5E3+%2B+ex%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >into L(p)and find out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28p%29+=+-4a-3bx-2cx%5E2-dx%5E3 ALIGN=MIDDLE  ALT=\"L%28p%29+=+-4a-3bx-2cx%5E2-dx%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "5.  If we let d = e = 0, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28a+%2Bbx%2B+cx%5E2+%2B+dx%5E3+%2B+ex%5E4%29=+L%28a+%2Bbx%2B+cx%5E2+%29+=+-4a-3bx-2cx%5E2 ALIGN=MIDDLE  ALT=\"L%28a+%2Bbx%2B+cx%5E2+%2B+dx%5E3+%2B+ex%5E4%29=+L%28a+%2Bbx%2B+cx%5E2+%29+=+-4a-3bx-2cx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Thus by inspection any polynomial of degree 2 will always be in the range of the map.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );