Select from the following sets, the one(s) which are linearlyindependent, given

Question

Select from the following sets, the one(s) which are linearlyindependent, given that each set is a subset of vectors from astandard vector space. C
{ [-7,-5,-5],    [0,-5,8],    [0,-8,0] } D
{ [-6,-9],    [-3,1],    [-5,8] } Select from the following sets, the one(s) which are linearlyindependent, given that each set is a subset of vectors from astandard vector space. C
{ [-7,-5,-5],    [0,-5,8],    [0,-8,0] } D
{ [-6,-9],    [-3,1],    [-5,8] } C
{ [-7,-5,-5],    [0,-5,8],    [0,-8,0] } D
{ [-6,-9],    [-3,1],    [-5,8] }

Explanation / Answer

First part; write the eqn in terms of x, y, z; Then; -7x-5y-5z=0........................1      -5y+8z=0........................2      -8y=0..............................3 for the sets to be linearly independent x=y=z=0 must only satisfythe eqn; then subtracting eqn 1 from 2 we have; 7x+13z=0 or, x=13z/7.................................4 from eqn 2 we have; z=5y/8; from eqn 3 we have; y=0; Therefore; x=y=z=0; Therefore the three sets are linearly imdependent. 2nd Part Similarly; Using linear combination method; a[-6,-9]+b[-3,1]=[-5,8] or, -6a-3b=-5..........................5 -9a+b=8..............................6 adding eqn 5 and 6; -33a=19; a=-19/33; b=8-(9*19/33) Therefore we have scalar values which satisfy linear combinationtherefore the sets are not linearly independent.

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