document.write( "Question 1159894: Find a basis for the subspace of R^3 consisting of all vectors [x1 x2 x3]
\n" ); document.write( "such that −2x1−7x2−4x3=0. \n" ); document.write( "

Algebra.Com's Answer #783051 by ikleyn(52788) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "Let  \"p\"  and  \"q\"  be two \"free\" parameters, i.e. arbitrary real numbers.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Consider the vectors  u = (-7p,2p,0)  and  v = (-2q,0,q).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "They both belong to the given subspace in .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Indeed, for  \"u\" :   -2*(-7p) - 7*(2p) - 4*0 = 14p - 14p - 0 = 0;\r\n" );
document.write( "\r\n" );
document.write( "    and     for  \"v\" :   -2*(-2q) - 7*0 - 4*q    =  4q -  0 - 4q = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Also, it is OBVIOUS that these vectors are LINEARLY INDEPENDENT.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, they form the basis in the given 2D subspace.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\"u\" and \"v\" are the desired vectors.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "/\/\/\/\/\/\/\/\/\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Hey, if you really are a student, learning Linear Algebra, can you post me the names of the books/textbooks, \r
\n" ); document.write( "\n" ); document.write( "recommended to you by your teacher/professor/lecturer ?\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "May be, I will be able to add my recommendations to your list.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );