document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1045619>Question 1045619</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Analyze the linear transformation L(x) = Dx that transforms the vector x = [1 4<BR>\n" );
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document.write( "vector L(x) = [3 6<BR>\n" );
document.write( "] and also transforms the vector x = [2 5<BR>\n" );
document.write( "] to the vector L(x) = [0 9<BR>\n" );
document.write( "] by <BR>\n" );
document.write( "doing the following:   <BR>\n" );
document.write( "1.  Find the entries of the 2x2 matrix D that result in this transformation.  2.  Describe, referencing the linear transformation, how the entries of matrix D were determined.  3.  Describe the relationship between the determinant of matrix D, the area of S, and the area of the transformed figure P by considering a 2x2 square S in the xy-plane.   <BR>\n" );
document.write( "Note: The linear transformation L(x) transforms the square S into another shape P in the xy-plane.    </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #661136 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=661136\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First of all, the format of your notation is a little awkward.<BR>\n" );
document.write( "If you're operating with row vectors then the linear transformation should be <BR>\n" );
document.write( "L(x) = xD, and not L(x) = Dx (which is used when x is a column vector.)\r<BR>\n" );
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document.write( "1. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+%28matrix%282%2C2%2C-5%2C2%2C2%2C1%29%29 ALIGN=MIDDLE  ALT=\"D+=+%28matrix%282%2C2%2C-5%2C2%2C2%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "2.  The entries of D were determined by first letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"D+=+%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and then making the following substitutions:\r<BR>\n" );
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document.write( "L(< 1,4 >) = < 1,4 >*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = < 3,6 >, and \r<BR>\n" );
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document.write( "L(< 2,5 >) = < 2,5 >*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = < 0,9 >.\r<BR>\n" );
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document.write( "The first equation gives the system a + 4c = 3 and b + 4d = 6.\r<BR>\n" );
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document.write( "The second equation gives the system 2a + 5c = 0 and 2b + 5d = 9.\r<BR>\n" );
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document.write( "Solving for a, b, c, and d simultaneously, we get a = -5, b = 2, c = 2, and d = 1.\r<BR>\n" );
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document.write( "3. Let S be the square with vertices (0,0), (0,2), (2,2), and (2,0).\r<BR>\n" );
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document.write( "S is sufficiently represented by the vectors < 0,2 > and < 2,0 > so it's enough to consider the effect of L on these two vectors.\r<BR>\n" );
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document.write( "L(< 0,2 >) = < 0,2 >*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = < 4,2 >, and\r<BR>\n" );
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document.write( "L(< 2,0 >) = < 2,0 >*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = < -10,4 >.\r<BR>\n" );
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document.write( "Hence the linear transformation L transforms the square S into a parallelogram P three of whose vertices are (-10,4), (0,0), <BR>\n" );
document.write( "and (4,2), and with an area of\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C1%2C-10%2C4%2C1%2C0%2C0%2C1%2C4%2C2%29%29+=+36 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C1%2C-10%2C4%2C1%2C0%2C0%2C1%2C4%2C2%29%29+=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now S has area 2*2 = 4, and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28D%29+=+abs%28abs%28matrix%282%2C2%2C-5%2C2%2C2%2C1%29%29%29+=+abs%28-9%29+=+9 ALIGN=MIDDLE  ALT=\"abs%28D%29+=+abs%28abs%28matrix%282%2C2%2C-5%2C2%2C2%2C1%29%29%29+=+abs%28-9%29+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get \r<BR>\n" );
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document.write( "36 = Area of P = |D|*(Area of S) = 9*4.\r<BR>\n" );
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