Finding vectors in the Span of v1 and v2

Question

Let v1=(2,−2,1) and v2=(2,−1,1).

Select all vectors below in the span of v1 and v2.

The answer choices are 
A. (0,−1,0) 
B. (0,0,0)
C. (2,−3,1)
D. (1,−3,0)

I couldn't understand how to solve this problem and which method to use. Can someone help me with this. I used the code filter to make it more understandable.

Answer 1

I will check answer A. and let you check the others:

Asking whether $(0,-1,0)$ is spanned by $v_1$ and $v_2$ is basically this question:

is there a solution to $(0,-1,0)=a(2,-2,1)+b(2,-1,1)$?

$2a+2b=0, -2a-b=-1, a+b=0$. The solution is $a=1, b=-1$.

(Let me know if you need help obtaining that solution.)

I.e., $(0,-1,0)=1(2,-2,1)-1(2,-1,1)$, so $(0,-1,0)$ is in the span of $v_1$ and $v_2$.

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Alternative solution using determinants:

$\begin{vmatrix}2&2&0\\-2&-1&-1\\1&1&0\end{vmatrix}=0,$

so $(2,-2,1), (2,-1,1)$ and $(0,-1,0)$ are linearly dependent;

i.e., $(0,-1,0)$ is in the span of $(2,-2,1)$ and $(2,-1,1)$.