[Math] Span of a set of vectors containing the zero vector

Question

I'm sorry if this is a stupid question, but if I have a set of $3$ vectors, one of which is the zero vector, and the question asks if the set of these three vectors spans $\mathbb R^2$, why is the answer yes?

I thought the three vectors had to be linearly independent to span $\mathbb R^2$, but if one of the vectors is the zero vector, isn't the set linearly dependent?

I even came across another question in the problem set asking to check if two vectors were linearly independent, and one of the two vectors was the zero vector and the answer was that they are not linearly independent. I'm just a bit confused.

Thanks!

Answer 1

if I have a set of 3 vectors, one of which is the zero vector, and the question asks if the set of these three vectors spans R2, why is the answer yes?

The answer is not necessarily yes. For example, consider $$ \{(0,0),(1,1),(2,2)\} $$

I thought the three vectors had to be linearly independent to span R2,

No. A set of two vectors must be linearly independent if it spans $\Bbb R^2$.

but if one of the vectors is the zero vector, isn't the set linearly dependent?

Yes, that's right.