document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=29839>Question 29839</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have to solve this proof but dont understand it all. <BR>\n" );
document.write( "Here it is: Let v and w be independent column vectors in R^3, and let A be an invertible 3X3 matrix.Prove that the vectors Av and Aw are independent.   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #16588 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=16588\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  You have to know a linear transformation T:V-->V  is invertible(one-to-one)<BR>\n" );
document.write( " <--> Ker(T)(null space of T) = {0}<BR>\n" );
document.write( " Let v and w be independent column vectors in R^3, and let A be <BR>\n" );
document.write( " an invertible 3X3 matrix.\r<BR>\n" );
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document.write( "Prove that the vectors Av and Aw are independent. <BR>\n" );
document.write( " Proof: A: R^3-->R^3 invertible <--> N(A) = {0} <--> rank A = 3.<BR>\n" );
document.write( " a Av + b Aw = 0 for two reals a, b<BR>\n" );
document.write( " --> A(av+bw) = 0<BR>\n" );
document.write( " --> av + bw belongs to N(A)= {0}<BR>\n" );
document.write( " --> av+vw = 0<BR>\n" );
document.write( " --> a=b = 0 (since v, w are lindep.)<BR>\n" );
document.write( " Hence, Av & Aw are l.indep.\r<BR>\n" );
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document.write( " This fact is true for any number of indep vectors.<BR>\n" );
document.write( " Also, it seems you have to work hard.\r<BR>\n" );
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document.write( " Kenny </SPAN>\n" );
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