Span of a set of vectors containing the zero vector

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!

• You seem to be confusing the span of a set of vectors with a basis for a vector space. A basis has to be linearly independent, but the span of a set of vectors is simply the set of all linear combinations of them. It doesn’t matter whether or not they’re linearly independent.
– amd
Dec 6, 2016 at 22:07

3 Answers

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.

• Right, so if I am given a set of four vectors, and asked if it spans R3, if three of those vectors are linearly independent it doesn't matter if the fourth one is isn't, the answer will be yes, it does span R3?
– melm
Dec 6, 2016 at 15:31
• That's exactly right. Dec 6, 2016 at 15:32
• Well, it's mostly right. It's not clear what you mean by "if the fourth one is isn't". A vector can't be "linearly independent/dependent" in its own right. However, the main idea is correct: if a set of three vectors span $\Bbb R^3$, then adding any other vector will yield another set that spans $\Bbb R^3$. Dec 6, 2016 at 15:35
• My issue was, in the question they set up a span of 4 vectors, and asked if it was "true" that they spanned R3. I found the RREF and found that there were 3 pivot columns. So I am wondering, by what you have said, that if 3 of the four vectors in the span are linearly independent (because there are 3 pivot columns), does that mean the span the question gave me, with all four vectors does span R3? Because what I learned in class was that when given a set of 4 vectors, all must be linearly independent in order to span R3 (which is impossible) which is why I was inclined to answer false.
– melm
Dec 6, 2016 at 15:41
• From what I learned in class, I would've thought that the span from the above question should only contain the three linearly independent vectors, not the fourth in order to span R3. Just wanted to know if my thinking is, well, very very wrong.
– melm
Dec 6, 2016 at 15:43

Since $\Bbb R^2$ has dimension $2$ so any two linearly independent vectors will span it.

No matter what the third vector be(even $0$) it will span $\Bbb R^2$ provided the other two vectors in the set are linearly independent

• I'm very confused, I also read that if the question gives me a set of four vectors and asks if spans R3, then the answer is no because only three vectors can span R3. How can any set span R2 if it has more than two vectors?
– melm
Dec 6, 2016 at 15:29
• A set of vectors will span a space if every vector in that space can be written a linear combination of them. In order for this to be possible the number of vectors in the set must be at least as big as the dimension of the space, but it can be bigger than the dimension. However, if it is bigger than the dimension the set will not be linearly independent. If the set is also linearly independent then that means that every vector in the space can be written UNIQUELY as a linear combination of those vectors in the set. Dec 6, 2016 at 15:48

A set of two vectors has to be linearly independent in order to span $\Bbb R^2$. If your set contains more vectors than that, then it is enough that you can find two linearly independent vectors among them.