document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=27385>Question 27385</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let V = matrix M(2×2)(Complex). Let W1 = {A = (Ajk) 1 <=j,k<=2 exists in V | A11 + A22 = 0 }.\r<BR>\n" );
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document.write( "a) Find subspaces W2 and W3 of V such that<BR>\n" );
document.write( "   V = W1 + W2, V = W1 + W3, W2 not equal to W3.\r<BR>\n" );
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document.write( "(Note: One way to describe a subspace is to give a basis for it.)\r<BR>\n" );
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document.write( "b) Explain why W2 (intersect) W3 = {0} for any pair of subspaces W2 and W3 satisfying the<BR>\n" );
document.write( "conditions of part a). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #15001 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=15001\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let V = matrix M(2×2)(Complex). Let W1 = {A = (Ajk) 1 <= j,k<= 2 exists in V | A11 + A22 = 0 }. <BR>\n" );
document.write( "a) Find subspaces W2 and W3 of V such that<BR>\n" );
document.write( "V = W1 + W2, V = W1 + W3, W2 not equal to W3. <BR>\n" );
document.write( "(Note: One way to describe a subspace is to give a basis for it.) \r<BR>\n" );
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document.write( "b) Explain why W2 (intersect) W3 = {0} for any pair of subspaces W2 and W3 satisfying the conditions of part a). \r<BR>\n" );
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document.write( "[ Note : Here W1+W2 should mean the direct sum instead of the general sum<BR>\n" );
document.write( " i.e W1 ^ W2 = {0})\r<BR>\n" );
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document.write( "Sol: a) Note dim V = 2x2 = 4 and since A11+A22 =0, W1 is a 3 dim subspace of V. By choosing a nonzero vector u , say u = I2 in V not in W1, set W2 to be generated by u, then we have W1 + W2 = V since dim W1 + W2 = 4. Similarly, choose w = <BR>\n" );
document.write( " (1 0)<BR>\n" );
document.write( " (0 0),  and let W3 be generated by w. We get W1 + W3 = V. Since u != cw for any scalar(complex), we see that W2, W3 are not equal. \r<BR>\n" );
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document.write( "b) If W2 & W3 are two distinct subspaces of V such that V = W1 + W2 = W1 + W3 (direct sum). <BR>\n" );
document.write( "Then dim W2 = dim W3 = 1. Claim: W2 ^ W3 = {0}. <BR>\n" );
document.write( " For otherwise, W2^W3 as a nonzero subspace of the 1-dim space W2 ( or W3),  then dim (W2 ^ W3) = 1 and so W2^W3 = W2 = W3. This contradicts to W2, W3 are not equal. Hence, W2 ^W3 must be {0}.\r<BR>\n" );
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document.write( " Kenny </SPAN>\n" );
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