document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=62609>Question 62609</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the linear transformation T : R3 -> R2 whose matrix <BR>\n" );
document.write( "with respect to the standard bases is given by <BR>\n" );
document.write( "2  1  0<BR>\n" );
document.write( "0  2 -1<BR>\n" );
document.write( "<BR>\n" );
document.write( "Now consider the bases:<BR>\n" );
document.write( "f1= (2, 4, 0)<BR>\n" );
document.write( "f2= (1, 0, 1)<BR>\n" );
document.write( "f3= (0, 3, 0) of R3 and\r<BR>\n" );
document.write( "\n" );
document.write( "g1= (1, 1)<BR>\n" );
document.write( "g2= (1,&#8722;1) of R2\r<BR>\n" );
document.write( "\n" );
document.write( "Compute the coordinate transformation matrices between the standard<BR>\n" );
document.write( "bases and these bases and compute the matrix of T with respect to the new<BR>\n" );
document.write( "bases.\r<BR>\n" );
document.write( "\n" );
document.write( "Any help would be appreciated. Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #43460 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=43460\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Consider the linear transformation T : R3 -> R2 whose matrix<BR>\n" );
document.write( "with respect to the standard bases is given by<BR>\n" );
document.write( "LET T=<BR>\n" );
document.write( "=<BR>\n" );
document.write( "2 1 0<BR>\n" );
document.write( "0 2 -1<BR>\n" );
document.write( "<BR>\n" );
document.write( "Now consider the bases:<BR>\n" );
document.write( "f1= (2, 4, 0)<BR>\n" );
document.write( "f2= (1, 0, 1)<BR>\n" );
document.write( "f3= (0, 3, 0) of R3 and<BR>\n" );
document.write( "g1= (1, 1)<BR>\n" );
document.write( "g2= (1,&#8722;1) of R2<BR>\n" );
document.write( "Compute the coordinate transformation matrices between the standard<BR>\n" );
document.write( "bases and these bases <BR>\n" );
document.write( "F IS THE MATRIX OF BASE VECTORS f1,f2,f3<BR>\n" );
document.write( "= <BR>\n" );
document.write( "2,1,0<BR>\n" );
document.write( "4,0,3<BR>\n" );
document.write( "0,1,0\r<BR>\n" );
document.write( "\n" );
document.write( "MATRIX OF STANDARD BASE IN R3 IS E3 SAY <BR>\n" );
document.write( "=<BR>\n" );
document.write( "1,0,0<BR>\n" );
document.write( "0,1,0<BR>\n" );
document.write( "0,0,1\r<BR>\n" );
document.write( "\n" );
document.write( "G IS THE MATRIX OF BASE VECTORS g1,g2.<BR>\n" );
document.write( "=<BR>\n" );
document.write( "1,1<BR>\n" );
document.write( "1,-1\r<BR>\n" );
document.write( "\n" );
document.write( "MATRIX OF STANDARD BASE IN R2 IS E2 SAY<BR>\n" );
document.write( "=<BR>\n" );
document.write( "1,0<BR>\n" );
document.write( "0,1\r<BR>\n" );
document.write( "\n" );
document.write( "F^(-1) IS [HOPE YOU KNOW HOW TO INVERT MATRIX.OTHERWISE PLEASE COME BACK]<BR>\n" );
document.write( "=<BR>\n" );
document.write( "0.5,0,-0.5<BR>\n" );
document.write( "0,  0,  1<BR>\n" );
document.write( "-2/3,1/3,2/3<BR>\n" );
document.write( "HENCE TRANSFORMATION MATRIX FROM E BASIS TO F BASIS IS<BR>\n" );
document.write( " (XF)=[F^(-1)]*(XE)  WHERE XF REPRESENTS X IN F BASIS.ETC..AND <BR>\n" );
document.write( "IF YE = A*(XE)......THEN<BR>\n" );
document.write( "(YF) = [F^-1]*(YE)\r<BR>\n" );
document.write( "\n" );
document.write( "G^-1 IS  <BR>\n" );
document.write( "=<BR>\n" );
document.write( "0.5,0.5<BR>\n" );
document.write( "0.5,-0.5<BR>\n" );
document.write( "SIMILARLY AS ABOVE,WE HAVE<BR>\n" );
document.write( "TRANSFORMATION MATRIX FROM E BASIS TO G BASIS IS<BR>\n" );
document.write( " (XG)=[G^(-1)]*(XE)  WHERE XG REPRESENTS X IN G BASIS.ETC..AND <BR>\n" );
document.write( "IF (YE) = A(XE)......THEN<BR>\n" );
document.write( "(YG) = [G^-1]*(YE)<BR>\n" );
document.write( "--------------------------------------------------------------------------<BR>\n" );
document.write( "and compute the matrix of T with respect to the new<BR>\n" );
document.write( "bases.<BR>\n" );
document.write( "HENCE <BR>\n" );
document.write( "IF TE IS IN E BASIS<BR>\n" );
document.write( "(TF)= [F^(-1)]*T*F<BR>\n" );
document.write( "TE is given <BR>\n" );
document.write( "=<BR>\n" );
document.write( "2,0<BR>\n" );
document.write( "1,2<BR>\n" );
document.write( "0,-1\r<BR>\n" );
document.write( "\n" );
document.write( "SIMILARLY IN G BASIS,TAKING T AS GIVEN, WE GET<BR>\n" );
document.write( "TG=[G^(-1)]*T*G\r<BR>\n" );
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document.write( "IT IS NOT CLEAR FROM YOUR PROBLEM WHETHER YOU ARE WRITING VETICAL VECTORS AS HORIZONTAL VECTORS FOR CONVENIENCE.PLEASE CLARIFY.IN SUCH A CASE BETTER WRITE AS (x1,x2,x3)'<BR>\n" );
document.write( "on your feed back we can complete the solution or you can continue using the above formula\r<BR>\n" );
document.write( "\n" );
document.write( "Any help would be appreciated. Thank you! </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );