document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=26683>Question 26683</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Verify that thesetof orthogonal nxn matrices form a subgroup of the general linear group GLn(R). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #14599 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=14599\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ORTHOGONAL MATRICES ARE SQUARE MATRICES SUCH THAT PRODUCT OF AN ORTHOGONAL MATRIX WITH ITS TRANSPOSE IS AN IDENTITY MATRIX.THAT IS IF<BR>\n" );
document.write( "A*A'=A'*A=I,THEN A IS AN ORTHOGONAL MATRIX.<BR>\n" );
document.write( "TO PROVE THAT THEY FORM A SUB GROUP UNDER THE GENERAL LINEAR GROUP OF MATRICES,WE NEED TO SHOW THAT<BR>\n" );
document.write( "I.ORTHOGONAL MATRICES ARE A SUB SET OF THE GENERAL GROUP OF MATRICES.<BR>\n" );
document.write( "II.UNDER THE SAME COMPOSITION AS IN THE GENERAL GROUP,THE SUB GROUP ITSELF IS A GROUP.HERE WE CONSIDER THE COMPOSITION OF MULTIPLICATION.THIS MEANS THAT WE HAVE TO SHOW WITHIN THE SUBGROUP FOR MULTIPLICATION......<BR>\n" );
document.write( "1.CLOSOURE<BR>\n" );
document.write( "2.ASSOCIATIVITY<BR>\n" );
document.write( "3.EXISTENCE OF IDENTITY<BR>\n" );
document.write( "4.EXISTENCE OF INVERSE <BR>\n" );
document.write( "NOW  I CRITERIA IS OBVIOUS AS THE SET OF ORTHOGONAL MATRICES (CALLED S) AS DEFINED ARE INDEED A SUB SET OF GENERAL LINEAR GROUP OF MATRICES(CALLED G).<BR>\n" );
document.write( "II....MULTIPLICATION IS INDEED WELL DEFINED AND EXISTS FOR ORTHOGONAL MATRICES AS IT IS FOR THE GENERAL GROUP OF MATRICES.<BR>\n" );
document.write( "1 AND 2.ASSOCIATIVITY HOLDS FOR MATRIX MULTIPLCATION IN GENERAL SO WE SHALL PROVE CLOSOURE HERE.<BR>\n" );
document.write( "LET  A AND B BE ORTHOGONAL MATRICES IN THE SUB SET OF ORTHOGONAL MATRICES S .SINCE A' AND B' ARE ALSO ORTHOGONAL MATRICES,THEY ARE ALSO ELEMENTS OF S.<BR>\n" );
document.write( "SO A*A'=A'*A=I...AND...B*B'=B'*B=I...<BR>\n" );
document.write( "NOW..<BR>\n" );
document.write( "IF WE SHOW THAT A*B IS ALSO AN ORTHOGONAL MATIX THEN IT WILL BE AN ELEMENT OF S WHICH PROVES CLOSOURE......<BR>\n" );
document.write( "TST (A*B)*(A*B)'=I<BR>\n" );
document.write( "WE KNOW FOR MATRIX MULTIPLICATION THAT (A*B)'=(B'*A')...HENCE<BR>\n" );
document.write( "(A*B)*(A*B)'=(A*B)*(B'*A')=A*(B*B')*A'=A*I*A'=A*A'=AA'=I<BR>\n" );
document.write( "HENCE A*B IS ALSO AN ORTHOGONAL MATRIX.HENCE IT IS AN ELEMENT OF S.<BR>\n" );
document.write( "3. IDENTITY ELEMENT IN S IS THE UNIT MATRIX WHICH IS SAME AS IN G.SINCE,WE HAVE <BR>\n" );
document.write( "A*I=I*A=A...WHICH FOLLOWS OBVIOUSLY FROM MULTIPLICATIVE PROPERTY.<BR>\n" );
document.write( "4.WE HAVE A*A'=A'*A=I.....IF A IS AN ORTHOGONAL MATRIX ITS INVERSE A' IS ALSO AN ORTHOGONAL MATRIX. HENCE IT IS AN ELEMENT OF S.<BR>\n" );
document.write( "THUS S IS A SUB GROUP OF G. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );