Check whether each of the following subsets of R³is linearly independent?

Question

i) #{[1,2,3],[−1,1,2],[2,1,1]}#
ii) #{[3,1,2],[−1,−1,−3],[−4,−3,0]}#

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Answer 1 · 2018-02-03T08:13:05

(i) : Linearly Dependent.

Part (i) :

Let us denote the given #R^3# vectors by,

#vecu=(1,2,3), vecv=(-1,1,2) and vecw=(2,1,1)#.

Then, these will be linearly independent,

#iff avecu+bvecv+cvecw=vec0; a,b,c in RRrArr a=b=c=0#.

#"Now, "avecu+bvecv+cvecw=vec0; a,b,c in RR#

#rArr(a,2a,3a)+(-b,b,2b)+(2c,c,c)=vec0=(0,0,0)#.

#(1) a-b+2c=0, (2) 2a+b+c=0, (3) 3a+2b+c=0#.

Solving #(1)-(3)" for "a,b,c," we get, "a=-1,b=1, c=1#.

Evidently, #{vecu, vecv, vec w}sub R^3# is linearly dependent.

Part (ii) can be similarly dealt with.