document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=250155>Question 250155</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have my final exam at 3 and I was not able to work through this problem on the study guide. \r<BR>\n" );
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document.write( "Let D: V->V be a linear map defined by D(v)=dv/dt. Find det(D) if V is the space generated by (e^t,e^2t,e^3t) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #182144 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=182144\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Step 1) Apply the linear map D to each vector in the basis <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left\{e^{t},e^{2t},e^{3t}\right\}\"> to get the vectors <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Et ALIGN=MIDDLE  ALT=\"e%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2e%5E%282t%29 ALIGN=MIDDLE  ALT=\"2e%5E%282t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3e%5E%283t%29 ALIGN=MIDDLE  ALT=\"3e%5E%283t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Step 2) Now let's write the output vectors <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Et ALIGN=MIDDLE  ALT=\"e%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2e%5E%282t%29 ALIGN=MIDDLE  ALT=\"2e%5E%282t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3e%5E%283t%29 ALIGN=MIDDLE  ALT=\"3e%5E%283t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as linear combinations of the basis vectors <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left\{e^{t},e^{2t},e^{3t}\right\}\">. \r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Et=1%2Ae%5Et%2B0%2Ae%5E%282t%29%2B0%2Ae%5E%283t%29 ALIGN=MIDDLE  ALT=\"e%5Et=1%2Ae%5Et%2B0%2Ae%5E%282t%29%2B0%2Ae%5E%283t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2e%5E%282t%29=0%2Ae%5Et%2B2%2Ae%5E%282t%29%2B0%2Ae%5E%283t%29 ALIGN=MIDDLE  ALT=\"2e%5E%282t%29=0%2Ae%5Et%2B2%2Ae%5E%282t%29%2B0%2Ae%5E%283t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3e%5E%283t%29=0%2Ae%5Et%2B0%2Ae%5E%282t%29%2B3%2Ae%5E%283t%29 ALIGN=MIDDLE  ALT=\"3e%5E%283t%29=0%2Ae%5Et%2B0%2Ae%5E%282t%29%2B3%2Ae%5E%283t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 3)\r<BR>\n" );
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document.write( "The coefficients to the linear combinations will form the matrix \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C2%2C0%2C0%2C0%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C2%2C0%2C0%2C0%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: the first column is formed from the coefficients in the first equation, the second from the second equation, etc.\r<BR>\n" );
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document.write( "Step 4) \r<BR>\n" );
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document.write( "Finding the determinant of a diagonal matrix is trivial since it is simply the product of the diagonal entries. So if \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C2%2C0%2C0%2C0%2C3%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C2%2C0%2C0%2C0%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=det%28A%29=1%2A2%2A3=6 ALIGN=MIDDLE  ALT=\"det%28A%29=1%2A2%2A3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Because the determinant of A is 6, the determinant of the linear transformation D is also 6. </SPAN>\n" );
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