document.write( "Question 28253: If the set {v1,v2,v3} of vectors of Rm is linearly dependent, then argue that the set {v1,v2,v3,v4} is also linearly dependent for every choice of v4 in Rm. \n" ); document.write( "
Algebra.Com's Answer #15370 by venugopalramana(3286)\"\" \"About 
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If the set {v1,v2,v3} of vectors of Rm is linearly dependent, then argue that the set {v1,v2,v3,v4} is also linearly dependent for every choice of v4 in Rm.\r
\n" ); document.write( "\n" ); document.write( "If the set {v1,v2,v3} of vectors of Rm is linearly dependent,
\n" ); document.write( "THAT IS WE HAVE A SET OF NUMBERS A1,A2,A3...NOT ALL EQUAL TO ZERO SUCH THAT
\n" ); document.write( "A1V1+A2V2+A3V3=0............I
\n" ); document.write( "WITHOUT LOSS OF ANY GENERALITY LET A1 BE NOT EQUAL TO ZERO.......II
\n" ); document.write( " then argue that the set {v1,v2,v3,v4} is also linearly dependent for every choice of v4 in Rm.
\n" ); document.write( "THAT IS TO SHOW THAT IF
\n" ); document.write( "B1V1+B2V2+B3V3+B4V4=0..THEN IT IMPLIES B1=B2=B3=B4=0 IS NOT NECSSARILY TRUE.
\n" ); document.write( "LET US TAKE B1=A1,B2=A2,B3=A3 AND B4=0..THEN WE HAVE
\n" ); document.write( "A1V1+A2V2+A3V3+0*V4=0+0=0...SINCE
\n" ); document.write( "A1V1+A2V2+A3V3=0.. AS PER EQN.I...BUT IN THIS WE HAVE
\n" ); document.write( "A1 IS NOT EQUAL TO ZERO AS PER II ABOVE.
\n" ); document.write( "HENCE the set {v1,v2,v3,v4} is also linearly dependent for every choice of v4 in Rm.\r
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