document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=106640>Question 106640</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Here T defined by   .Find a vector “x” whose image under T is b, and determines whether x is unique.<BR>\n" );
document.write( "A=1  -3  2<BR>\n" );
document.write( "  0  1  -4<BR>\n" );
document.write( "  3  -5 -9<BR>\n" );
document.write( "b= 6<BR>\n" );
document.write( "  -7<BR>\n" );
document.write( "  -9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #77649 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=77649\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T:R^n\rightarrow R^m\"> such that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T(\v{x})=A\v{x}\">\r<BR>\n" );
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document.write( "Remember <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE T(\v{x})=\v{b}\">, so this means <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A\v{x}=\v{b}\">\r<BR>\n" );
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document.write( "Now plug in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A=\begin{bmatrix} 1 & -3 &  2 \\ 0 & 1 & -4 \\ 3 & -5 & -9 \end{bmatrix}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \v{b}=\begin{bmatrix} 6\\ -7 \\ -9 \end{bmatrix}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \begin{bmatrix} 1 & -3 &  2 \\ 0 & 1 & -4 \\ 3 & -5 & -9 \end{bmatrix}\v{x}=\begin{bmatrix} 6\\ -7 \\ -9 \end{bmatrix}\">\r<BR>\n" );
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document.write( "So which vectors <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{x}\"> will satisfy this matrix vector equation?\r<BR>\n" );
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document.write( "To find out, simply set up the augmented matrix <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left[\textrm{A \v{b}}\right]\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \begin{bmatrix} 1 & -3 &  2 & 6\\ 0 & 1 & -4 & -7\\ 3 & -5 & -9 & -9\end{bmatrix}\">\r<BR>\n" );
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document.write( "Now row reduce the augmented matrix to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \begin{bmatrix} 1 & 0 &  0 & -5\\ 0 & 1 & 0 & -3\\ 0 & 0 & 1 & 1 \end{bmatrix}\">\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{x}=\begin{bmatrix} -5\\ -3\\ 1 \end{bmatrix}\"> which means <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{x}\"> is unique\r<BR>\n" );
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document.write( " </SPAN>\n" );
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