document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=3698>Question 3698</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.	Find bases for the following subspaces of F5:<BR>\n" );
document.write( "W1 = {(a1, a2, a3, a4, a5) Î F5: a1 – a3 – a4 = 0} and<BR>\n" );
document.write( "W2 = {(a1, a2, a3, a4, a5) Î F5: a2 = a3 = a4 and a1 + a5 = 0}.<BR>\n" );
document.write( "What are the dimensions of W1 and W2?<BR>\n" );
document.write( "2.	The set of all upper triangular n x n matrices is a subspace W of<BR>\n" );
document.write( "M n x n (F). Find a basis for W and determine its dimension.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #1640 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=1640\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  1. Sol:   <BR>\n" );
document.write( "         About W1, since a1 – a3 – a4 = 0, or a1 = a3 + a4.<BR>\n" );
document.write( "         We see that {(s+t,r,s,t,u)|r,s,t,u in F} is a general<BR>\n" );
document.write( "         form of vectors in W1.\r<BR>\n" );
document.write( "\n" );
document.write( "         Choose a1=1, a3=-a4 = 1 and then a4=0, a1=a3=1<BR>\n" );
document.write( "         we see that (1,a2,1,-1,a5), (1,a2,1,0,a5) are linear independent<BR>\n" );
document.write( "         of W1 for all scalars a2,a5 in F.<BR>\n" );
document.write( "         Thus, {(1,1,1,-1,0), (1,0,1,-1,1),(1,1,1,0,0),(1,0,1,0,1)}<BR>\n" );
document.write( "         is a set of 4 independent vectors.<BR>\n" );
document.write( "         Also, note W1 is a proper subspace of F5 and hence dimW1 <=4.<BR>\n" );
document.write( "        We claim that {(1,1,1,-1,0), (1,0,1,-1,1),(1,1,1,0,0),(1,0,1,0,1)}      <BR>\n" );
document.write( "        a basis of W4 and hence Dim W1 = 4.<BR>\n" );
document.write( " <BR>\n" );
document.write( "         For W2, since a2 = a3, a3 = a4 and a1 + a5 = 0.<BR>\n" );
document.write( "         We see that {(s,t,t,t,-s)|s,t in F} is a general<BR>\n" );
document.write( "         form of vectors in W2. <BR>\n" );
document.write( "         Clearly, dim W2 = 2 from this general form.<BR>\n" );
document.write( "         By choosing a2=a3 = 1, a3=a4=1, and a1=a5 = 0.<BR>\n" );
document.write( "         And, choose a2=a3=a4 =0, and a1= -a5=1.<BR>\n" );
document.write( "         We obtain two independent vectors (1,1,1,0,0) and<BR>\n" );
document.write( "         (1,0,0,0,-1) in W2. <BR>\n" );
document.write( "         Thus, {(1,1,1,0,0),(1,0,0,0,-1)} is a basis of W2 and<BR>\n" );
document.write( "         dim W2 = 2.\r<BR>\n" );
document.write( "\n" );
document.write( " 2. Note the dimension of M n x n (F) isn nxn = n^2.<BR>\n" );
document.write( "    There are (1+2+3+..+n) = n(n+1)/2 free entries in<BR>\n" );
document.write( "    an upper upper triangular of M n x n (F).<BR>\n" );
document.write( "    Hence, dim W = n(n+1)/2.\r<BR>\n" );
document.write( "\n" );
document.write( "         [Discussion of question 1:<BR>\n" );
document.write( "         First of all , try to find the dimensions of W1 and W2.<BR>\n" );
document.write( "         Since the restraint of W1 is the linear equation <BR>\n" );
document.write( "         a1 – a3 – a4 = 0, so dim W1 = 5 -1 = 4.<BR>\n" );
document.write( "         For W2, there are three linear restraints, namely,  <BR>\n" );
document.write( "         a2 = a3, a3 = a4 and a1 + a5 = 0<BR>\n" );
document.write( "         if they are independent,then dim W2 = 5 -3 = 2. <BR>\n" );
document.write( "         Since, clearly,two of the 3 equations are independent,<BR>\n" );
document.write( "         we see that dim W2 = 2 or 3. ]\r<BR>\n" );
document.write( "\n" );
document.write( " Kenny<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );