document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1094593>Question 1094593</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that there exists a linear transformation T : R2 ! R3 such that T(1; 1) = (1; 0; 2) and<BR>\n" );
document.write( "T(2; 3) = (1;-1; 4): What is T(8; 11)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #709245 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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/cgi-bin/show-face.mpl?user=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=709245\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let V = {(1; 1), (2; 3)}<BR>\n" );
document.write( "let U = {(1 0;2), (1; -1; 4)}<BR>\n" );
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document.write( "check if V is a linearly independent or dependent set of vectors<BR>\n" );
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document.write( "write the vectors of V as a matrix with each vector written as a column<BR>\n" );
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document.write( "| 1 2 |<BR>\n" );
document.write( "| 1 3 |<BR>\n" );
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document.write( "transform the matrix to row echelon form<BR>\n" );
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document.write( "r2 - r1 ---> r2<BR>\n" );
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document.write( "| 1 2 | <BR>\n" );
document.write( "| 0 1 |<BR>\n" );
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document.write( "r1 - 2r2 ---> r1<BR>\n" );
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document.write( "| 1 0 |<BR>\n" );
document.write( "| 0 1 |<BR>\n" );
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document.write( "Rank is 2 implies the vectors are linearly independent, furthermore any set of two linearly independent vectors in R2 spans R2. Hence V is a basis for R2.<BR>\n" );
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document.write( "Since V is a basis, there exists only one linear transformation that maps the vectors of V into the vectors of U.<BR>\n" );
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document.write( "To solve this case, multiply the matrix composed by the vectors of U as columns, by the inverse of the matrix composed by the vectors of V as columns.<BR>\n" );
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document.write( "Note that since the vectors of V form a basis for R2, V is invertible. <BR>\n" );
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document.write( "We invert the following matrix<BR>\n" );
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document.write( "| 1 2 |<BR>\n" );
document.write( "| 1 3 |<BR>\n" );
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document.write( "determinant of this matrix = 3 - 2 = 1, and the inverse matrix is<BR>\n" );
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document.write( "| 3 -2 |<BR>\n" );
document.write( "| -1 1 |<BR>\n" );
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document.write( "| 1 1  |  times | 3 -2 |<BR>\n" );
document.write( "| 0 -1 |        | -1 1 |<BR>\n" );
document.write( "| 2 4  |<BR>\n" );
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document.write( "result is<BR>\n" );
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document.write( "| 2 -1 |<BR>\n" );
document.write( "| 1 -1 |<BR>\n" );
document.write( "| 2  0 |<BR>\n" );
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document.write( "this is our linear transformation T<BR>\n" );
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document.write( "you can check that T maps the given R2 vectors to the vectors in U, you do this by multiplying our T matrix by the vector from V<BR>\n" );
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document.write( "we follow this procedure for (8, 11) and we get<BR>\n" );
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document.write( "(5, -3, 16)<BR>\n" );
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document.write( " </SPAN>\n" );
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