Let S={v1=(2,2,3), v2=(-1,-2,1), v3=(0,1,0)}. Find a condition on a,b, and c so that v=(a,b,c) is a linear combination of v1,v2 and v3?

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Answer 1 · 2016-12-17T15:21:03

See below.

#v_1,v_2# and #v_3# span #RR^3# because

#det({v_1,v_2,v_3})=-5 ne 0#

so, any vector #v in RR^3# can be generated as a linear combination of #v_1,v_2# and #v_3#

The condition is

#((a),(b),(c))= lambda_1 ((2),(2),(3))+lambda_2((-1),(-2),(1))+lambda_3((0),(1),(0))# equivalent to the linear system

#((2,-1,0),(2,-2,1),(3,1,0))((lambda_1),(lambda_2),(lambda_3))=((a),(b),(c))#

Solving for #lambda_1,lambda_2,lambda_3# we will have the #v# components in the reference #v_1,v_2,v_2#