document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=4563>Question 4563</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let T, U: V to W be linear transformations. Prove that:<BR>\n" );
document.write( "a. R(T+U) is a subset of R(T)+R(U).<BR>\n" );
document.write( "b. If W is finite-dimensional, then rank(T+U)=< rank(T)+rank(U).<BR>\n" );
document.write( "c. Deduct from b that rank(A+B)=< rank(A)+rank(B) for any m x n matrices<BR>\n" );
document.write( "A and B. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #2096 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=2096\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Let T, U: V to W be linear transformations. Prove that:<BR>\n" );
document.write( "a. R(T+U) is a subset of R(T)+R(U).<BR>\n" );
document.write( "b. If W is finite-dimensional, then rank(T+U)=< rank(T)+rank(U).<BR>\n" );
document.write( "c. Deduct from b that rank(A+B)=< rank(A)+rank(B) for any m x n matrices<BR>\n" );
document.write( "A and B.\r<BR>\n" );
document.write( "\n" );
document.write( " Proof: a) x',y' are in R(T+U)<BR>\n" );
document.write( "       --> there exist x,y in V such that (T+U)(x) = x', (T+U)(y) = y'<BR>\n" );
document.write( "       --> x'+y'=(T+U)(x)+(T+U)(y)=(T+U)(x+y)= T(x)+U(x) + T(y)+ U(y)<BR>\n" );
document.write( "           = T(x+y) + U(x+y) <BR>\n" );
document.write( "       ---> x'+y' is in R(T)+R(U)<BR>\n" );
document.write( "        x' is in R(T+U) and c is a scalar<BR>\n" );
document.write( "      --> there exist x,y in V such that (T+U)(x) = x'<BR>\n" );
document.write( "      ---> (T+U)(cx) = c(T+U)(x) = cx' = T(cx)+ U(cx) for some x in V<BR>\n" );
document.write( "      --> cx' is in R(T) + R(U)<BR>\n" );
document.write( "      This shows R(T+U) is a subspace of R(T)+R(U).\r<BR>\n" );
document.write( "\n" );
document.write( "    b) Note T+U: V-->W is a linear transformation,<BR>\n" );
document.write( "       rank(T) and rank(U) <= dim W (finite)<BR>\n" );
document.write( "       Hence, rank(T) + rank(U) is finite.<BR>\n" );
document.write( "       By a) R(T+U) is a subspace of R(T)+R(U),<BR>\n" );
document.write( "       let W' be the intersection of R(T) and R(U).<BR>\n" );
document.write( "       we have dim R(T) + dim R(U) = dim [R(T)+R(U)]+ dim(W')<BR>\n" );
document.write( "       that is, rank(T) + rank(U) = rank(T+U) + dim(W')<BR>\n" );
document.write( "       Hence, rank(T+U) <= rank(T)+rank(U)\r<BR>\n" );
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document.write( "   c) Consider<BR>\n" );
document.write( "      Define two linear transformations T, U from R^n to R^m<BR>\n" );
document.write( "      (or from F^n to F^m, column vectors of size n or m)<BR>\n" );
document.write( "      by T(ei) = E Aij fj   and U(ei) = E Bij fj,<BR>\n" );
document.write( "    (where E means sumation){ei+i=1,,..,n} {fj|j=1,2,..m} are<BR>\n" );
document.write( "    standard basis of R^n & R^m resp.<BR>\n" );
document.write( "    Since rank(T) = rank(A) and rank(U) = rank(B)<BR>\n" );
document.write( "      By b), we have rank(T+U) <= rank(T)+rank(U)<BR>\n" );
document.write( "   Hence , rank(A+B)<= rank(A)+rank(B) for any m x n matrices<BR>\n" );
document.write( "   A and B.<BR>\n" );
document.write( " <BR>\n" );
document.write( " Kenny<BR>\n" );
document.write( " PS. You have to use the theorems in the text tofill some details. <BR>\n" );
document.write( " In a) R(T+U) should be a subspace ofR(T)+R(U) , <BR>\n" );
document.write( "  as a subset does not make much sense<BR>\n" );
document.write( " <BR>\n" );
document.write( " <BR>\n" );
document.write( "        </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );