How to tell if Linearly Independent

Question

just a quick question. Do 2 vectors in R3 mean it is linearly independent? The question " is [1 2 3], [0 2 3] linearly independent?" I solved it by saying there is no scalar multiple thus linearly independent. But I was wondering if 2 vectors in R3 also mean linearly independent.

Thanks in advance

Answer 1

No, for example $(1,0,0)$ and $(2,0,0)$ are not linearly independent.

Answer 2

Note that for

• $v_1=(1,2,3)$
• $v_2=(0,2,3)$

we have that

• $v_3=v_1-v_2=(1,0,0)=e_1$
• $v_2=(0,2,3)=2e_2+3e_3$

thus $v_3$ and $v_2$ are linearly independent and thus also $v_1$ and $v_2$ are linearly independent.

The standard method to check in general linear independence is to arrange vectors in a matrix and determine the rank by RREF or determinants.