document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=15127>Question 15127</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A^T denotes the transpose of a matrix A.<BR>\n" );
document.write( "Show that <A,B> = Tr(B^T,A) defines an inner product on R^(n x n)\r<BR>\n" );
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document.write( "Let U be the set of symmetric 2 x 2 matrices with real entries. U is subspace of R (2x2). Find an orthonormal basis of U with respect to the above inner product. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #7533 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=7533\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  A : nxn square matrix over R. (missing)<BR>\n" );
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document.write( "A^T denotes the transpose of a matrix A.<BR>\n" );
document.write( "Show that = Tr(B^T,A) defines an inner product on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E%28n+x+n%29+ ALIGN=MIDDLE  ALT=\"R%5E%28n+x+n%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let U be the set of symmetric 2 x 2 matrices with real entries. U is subspace of R (2x2). Find an orthonormal basis of U with respect to the above inner product.\r<BR>\n" );
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document.write( " Let A = (aij), B = (bij) be nxn sq. matrices over R. (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SIGMA ALIGN=MIDDLE  ALT=\"SIGMA\"   DISABLED_style= \"max-width:90%; height:auto;\" > means summation)<BR>\n" );
document.write( " Define (A,B) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Tr%28B%5E%28T%29+A%29+ ALIGN=MIDDLE  ALT=\"Tr%28B%5E%28T%29+A%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+SIGMA+ ALIGN=MIDDLE  ALT=\"+SIGMA+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SIGMA ALIGN=MIDDLE  ALT=\"SIGMA\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5Bki%5D+a%5Bki%5D+ ALIGN=MIDDLE  ALT=\"+b%5Bki%5D+a%5Bki%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" >(k=1,2..,n)(i=1,2..,n)<BR>\n" );
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document.write( " You have to check the definition of inner product (as symmetric<BR>\n" );
document.write( " , linear, positive definite, all very easy)<BR>\n" );
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document.write( " Note the dim of the vector space <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E%282x2%29 ALIGN=MIDDLE  ALT=\"R%5E%282x2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 4.<BR>\n" );
document.write( " And dim U = 3 (why?)<BR>\n" );
document.write( " Let A=<BR>\n" );
document.write( " (1 0)<BR>\n" );
document.write( " (0 0)<BR>\n" );
document.write( " B=<BR>\n" );
document.write( " (0 0)<BR>\n" );
document.write( " (0 1)<BR>\n" );
document.write( " C =<BR>\n" );
document.write( " (0 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%282%29+ ALIGN=MIDDLE  ALT=\"1%2Fsqrt%282%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( " (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"1%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 0)<BR>\n" );
document.write( " then {A,B,C} forms an o.n. basis of U.\r<BR>\n" );
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document.write( " You should test (A,A)=(B,B)=(C,C)= 1.<BR>\n" );
document.write( "  (A,B) = (B,C) =(C,A) = 0\r<BR>\n" );
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document.write( " Kenny<BR>\n" );
document.write( " </SPAN>\n" );
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