For the following set of vectors, determine whether it is linearly independent or linearly dependent. If it is linearly dependent, find all its maximum linearly independent subsets? {(1,2,-1),(2,4,6),(0,0,-8)}

Question

from my ERO, I found out that it is linearly dependent but I dont know how to find its linearly independent subsets

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Answer 1 · 2018-07-09T09:59:41

Please see the explanation below.

The vectors are

#{(1,2,-1),(2,4,6),(0,0,-8)}#

The vectors are independent if

#alpha((1),(2),(-1))+beta((2),(4),(6))+gamma((0),(0),(-8))=((0),(0),(0))#

Where #alpha, beta, gamma in RR^3#

has only the trivial solution

#alpha=beta=gamma=0#

Perform a row reduction on the augmented matrix

#A=((1,2,0,|,0),(2,4,0,|,0),(-1,6,-8, |,0))#

#<=>#, #R2larr R2-2R1#

#((1,2,0,|,0),(0,0,0,|,0),(-1,6,-8, |,0))#

#<=>#, #R2harr R3#

#((1,2,0,|,0),(-1,6,-8,|,0),(0,0,0, |,0))#

#<=>#, #R2larr R2+R1#

#((1,2,0,|,0),(0,8,-8,|,0),(0,0,0, |,0))#

#<=>#, #R2larr (R2)/8#

#((1,2,0,|,0),(0,1,-1,|,0),(0,0,0, |,0))#

#<=>#, #R1larr R1-2R2#

#((1,0,2,|,0),(0,1,-1,|,0),(0,0,0, |,0))#

Since there is a free variable, the system is not linearly independant

The determinant of the matrix #=0#

Therefore,

#{(alpha=-2gamma),(beta=gamma),(gamma= " free"):}#

So,

#-2gamma((1),(2),(-1))+gamma((2),(4),(6))+gamma((0),(0),(-8))=((0),(0),(0))#

Hope that this will help!!!