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 $v_1,v_2,v_2$