document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=5373>Question 5373</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let T be a linear operator on a finite-dimensional vector space V, and let<BR>\n" );
document.write( "b (beta) be an ordered basis for V. Prove that l (lamda) is an eigenvalue of T if and only if l (lamda) is an eigenvalue of [T]b (beta). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #2727 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=2727\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Let T be a linear operator on a finite-dimensional vector space V, and let<BR>\n" );
document.write( "b (beta) be an ordered basis for V. Prove that l (lamda) is an eigenvalue of T if and only if l (lamda) is an eigenvalue of [T]b (beta).\r<BR>\n" );
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document.write( " ==>  if lamda is an eigenvalue of T , then there exists nonzero vector <BR>\n" );
document.write( "      x in V such that Tx = lambda x,<BR>\n" );
document.write( "      Since B(better using capital letter for basis) is an order basis<BR>\n" );
document.write( "      of V, let B ={v1,v2,..,vn} (assume dim  V = n)<BR>\n" );
document.write( "      x = E aivi for scalars a1,a2,..,an (E means summation over i)<BR>\n" );
document.write( "      i.e. xB = (a1,a2,..,an)^T (T means transpose) be a column vector(in F^n)<BR>\n" );
document.write( "      Let [T]B = [Tij](nxn matrix)<BR>\n" );
document.write( "      Since Tx = lambda x, expressing in the o.b. B, we<BR>\n" );
document.write( "      have [Tx]B = [lambda x]B, <BR>\n" );
document.write( "   So, we get [Tx]B =[T]B xB = lambda xB<BR>\n" );
document.write( "   Also, x is nonzero vector implies xB is non-zero in F^n.<BR>\n" );
document.write( "   This shows lambda is an eigenvector of [T]B.<BR>\n" );
document.write( " <== If lambda is an eigenvector of [T]B , then<BR>\n" );
document.write( "     there exists a non-zero column vector w=(c1,c2,..,cn)^T in F^n<BR>\n" );
document.write( "      such that [Tx]B w = lambda w.<BR>\n" );
document.write( "     Note B = {v1,v2,..,vn} and set x = E civi<BR>\n" );
document.write( "     then we have xB = w and <BR>\n" );
document.write( "     [Tx]B = [T]B xB = [T]B w = lambda w = lambda xB,<BR>\n" );
document.write( "     This implies Tx = lambda x because B is a basis of V.<BR>\n" );
document.write( "   (a linear operator isnquely determined by the values on a basis)<BR>\n" );
document.write( "     Also, x is non-zero since w is non-zero.<BR>\n" );
document.write( "     This proves lambda is an eigenvalue of T.\r<BR>\n" );
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document.write( " Kenny\r<BR>\n" );
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document.write( " PS:<BR>\n" );
document.write( "  By definition of [T]B = [Tij] (nxn matrix) <BR>\n" );
document.write( "  if B = {v1,v2,..,vn}<BR>\n" );
document.write( "   for each i,T (vi) = E Tij vj (summation over j)<BR>\n" );
document.write( "    </SPAN>\n" );
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