document.write( "Question 1160970: Let v=[4, -9, 1, 9]. Find a basis of the subspace of R^4 consisting of all vectors perpendicular to v. \n" ); document.write( "

Algebra.Com's Answer #785166 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "            The other tutor retold the general theory, but the meaning of the problem and the meaning of the request
\n" ); document.write( "            is to point (to find) three concrete linearly independent vectors in $\"R%5E4\"$ perpendicular to vector V.\r
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\n" ); document.write( "\n" ); document.write( "            It was not done in the post by the other tutor, so I will do it right now.\r
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document.write( "The first vector in , perpendicular to vector v, is  x = (1,0,-4,0).\r\n" );
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document.write( "    Indeed, you can check it on your own that the scalar product of vectors v and x is equal to zero\r\n" );
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document.write( "        (v,x) = 4*1 + (-9)*0 + 1*(-4) + 9*0 = 4 + 0 - 4 + 0 = 0.\r\n" );
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document.write( "The second vector in , perpendicular to vector v, is  y = (0,-1,-9,0).\r\n" );
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document.write( "    Indeed, you can check it on your own that the scalar product of vectors v and y is equal to zero\r\n" );
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document.write( "        (v,y) = 4*0 + (-9)*(-1) + 1*(-9) + 9*0 = 0 + 9 - 9 + 0 = 0.\r\n" );
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document.write( "Finally, the third vector in , perpendicular to vector v, is  z = (0,0,-9,1).\r\n" );
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document.write( "    Indeed, you can check it on your own that the scalar product of vectors v and z is equal to zero\r\n" );
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document.write( "        (v,z) = 4*0 + (-9)*0 + 1*(-9) + 9*1 = 0 + 0 - 9 + 9 = 0.\r\n" );
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document.write( "Next, it is OBVIOUS that the three vectors\r\n" );
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document.write( "        x = (1, 0, -4, 0),\r\n" );
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document.write( "        y = (0,-1, -9, 0)  and\r\n" );
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document.write( "        z = (0, 0, -9, 1)\r\n" );
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document.write( "are linearly independent (due to construction of their components).\r\n" );
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document.write( "Thus we constructed (guessed, based on intuition) three linearly independent vectors in  perpendicular to the given vector v.\r\n" );
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document.write( "Hence, these three vectors x, y and z form a basis in the orthogonal complement to vector V in .\r\n" );
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document.write( "It is what has to be done.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved, explained and completed.\r
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