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.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! 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.
If the set {v1,v2,v3} of vectors of Rm is linearly dependent,
THAT IS WE HAVE A SET OF NUMBERS A1,A2,A3...NOT ALL EQUAL TO ZERO SUCH THAT
A1V1+A2V2+A3V3=0............I
WITHOUT LOSS OF ANY GENERALITY LET A1 BE NOT EQUAL TO ZERO.......II
then argue that the set {v1,v2,v3,v4} is also linearly dependent for every choice of v4 in Rm.
THAT IS TO SHOW THAT IF
B1V1+B2V2+B3V3+B4V4=0..THEN IT IMPLIES B1=B2=B3=B4=0 IS NOT NECSSARILY TRUE.
LET US TAKE B1=A1,B2=A2,B3=A3 AND B4=0..THEN WE HAVE
A1V1+A2V2+A3V3+0*V4=0+0=0...SINCE
A1V1+A2V2+A3V3=0.. AS PER EQN.I...BUT IN THIS WE HAVE
A1 IS NOT EQUAL TO ZERO AS PER II ABOVE.
HENCE the set {v1,v2,v3,v4} is also linearly dependent for every choice of v4 in Rm.
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