Question

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?

Answer 1

See below.

Explanation:

`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`