document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=375894>Question 375894</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let U be a subspace of R^4 and let the set S={x1,x2,x3} be an orthogonal basis of U, where x1=[1,0,-1,-1]  x2=[2,1,1,1]  x3=[-1,3,-1,0]\r<BR>\n" );
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document.write( "a) find a basis of U^(orthogonal symbol)\r<BR>\n" );
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document.write( "b) Given x=[3,1,0,42] find vectors x1 and x2 such that x= x1+x2, where x1 is in U and x2 is in U^(orthogonal symbol) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #267439 by </B><A HREF=/tutors/aboutme.mpl?userid=Jk22><B>Jk22(389)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Jk22><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=Jk22><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=267439\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) We take a vector which is orthogonal to : x1=[1,0,-1,-1]  x2=[2,1,1,1]  x3=[-1,3,-1,0]\r<BR>\n" );
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document.write( "let u[a,b,c,d], u.x1=u.x2=u.x3=0<BR>\n" );
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document.write( "a-c-d=0<BR>\n" );
document.write( "2a+b+c+d=0<BR>\n" );
document.write( "-a+3b-c=0\r<BR>\n" );
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document.write( "-a+3b-c=0<BR>\n" );
document.write( "3b-2c-d=0<BR>\n" );
document.write( "7b-c+d=0\r<BR>\n" );
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document.write( "-a+3b-c=0<BR>\n" );
document.write( "3b-2c+d=0<BR>\n" );
document.write( "10b-3c=0<BR>\n" );
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document.write( "3c=10b<BR>\n" );
document.write( "d=3b-2*10b/3=-11b/3<BR>\n" );
document.write( "a=3b-c=3b-10b/3=-b/3\r<BR>\n" );
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document.write( "u=[-1,3,10,-11]<BR>\n" );
document.write( " \r<BR>\n" );
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document.write( " \r<BR>\n" );
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document.write( "since it's of dimension 1 (x1,x2,x3 are linearly independent), another way would be to calculate the cross product of x1,x2,x3 : \r<BR>\n" );
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document.write( "[e1 1 2 -1]<BR>\n" );
document.write( "[e2 0 1 3]<BR>\n" );
document.write( "[e3 -1 1 -1]<BR>\n" );
document.write( "[e4 -1 1 0]\r<BR>\n" );
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document.write( "=e1<BR>\n" );
document.write( "[0 1 3]<BR>\n" );
document.write( "[-1 1 -1]<BR>\n" );
document.write( "[-1 1 0]<BR>\n" );
document.write( " <BR>\n" );
document.write( "-e2<BR>\n" );
document.write( "[1 2 -1]<BR>\n" );
document.write( "[-1 1 -1]<BR>\n" );
document.write( "[-1 1 0]<BR>\n" );
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document.write( "+e3<BR>\n" );
document.write( "[1 2 -1]<BR>\n" );
document.write( "[0 1 3]<BR>\n" );
document.write( "[-1 1 0]<BR>\n" );
document.write( " <BR>\n" );
document.write( "-e4<BR>\n" );
document.write( "[1 2 -1]<BR>\n" );
document.write( "[0 1 3]<BR>\n" );
document.write( "[-1 1 -1]<BR>\n" );
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document.write( "=e1(-(-1)*(1*0-1*3)-1*(1*(-1-3)))<BR>\n" );
document.write( "-e2(-1*(-2+1)-1*(-1-1))<BR>\n" );
document.write( "+e3(-3-7)<BR>\n" );
document.write( "-e4(-4-(7))<BR>\n" );
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document.write( "=[1,-3,-10,11]=u1=-u\r<BR>\n" );
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document.write( "b) Given x=[3,1,0,42] find vectors x1 and x2 such that x= x1+x2, where x1 is in U and x2 is in U^(orthogonal symbol)\r<BR>\n" );
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document.write( "we project x on u1 : \r<BR>\n" );
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document.write( "fact=x.u1/|u1|^2=(3-3+42*11)/Sqrt(1+9+100+121)=462/231=2\r<BR>\n" );
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document.write( "x2=fact*u1=[1,-3,-10,11]*2=[2,-6,-20,22]<BR>\n" );
document.write( "x1=x-x2=[3,1,0,42]-[2,-6,-20,22]=[1,7,20,20]<BR>\n" );
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