How to determine if these three vectors are linearly independent or linearly dependent?

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Answer 1 · 2018-04-17T05:12:43

The vectors are lin. ind.

By definition, the given vectors will be lin. ind., if and only if,

#x,y,z in RR; xvecv_1+yvecv_2+zvecv_3=vec0 rArr x=y=z=0#.

Now, #x,y,z in RR; xvecv_1+yvecv_2+zvecv_3=vec0#.

# rArr x[(3),(-4),(-3)]+y[(-4),(-5),(-2)]+z[(0),(-2),(-5)]=[(0),(0),(0)]#.

#rArr [(3x),(-4x),(-3x)]+[(-4y),(-5y),(-2y)]+[(0),(-2z),(-5z)]=[(0),(0),(0)]#.

#rArr [(3x-4y),(-4x-5y-2z),(-3x-2y-5z)]=[(0),(0),(0)]#.

#rArr 3x-4y=4x+5y+2z=3x+2y+5z=0........(lambda)#.

Solving the #(lambda)# system of eqns. for #x,y,z#, we get,

#x=y=z=0#.

Consequently, the vectors are lin. ind.