Is the zero vector always in the span of any set of vectors?

[QuarkCharmer]

Homework Statement

Homework Equations

The Attempt at a Solution

It's not so much a homework problem as it is something I was wondering. Our book is horrible, and does not explicitly state that the zero vector is always in the span of two vectors. If I am understanding things right:

if v and u are vectors
$$span(v, u)$$
is the collection of all points that can be reached via a linear combination of v and u. My reasoning is that if v is equal to u, then span{v,u} = span{v} = span{u}, which is essentially a line. However, it seems to me that in any space, R^2, R^3,...,R^n, the span{} of any n vectors will always go through the origin and thus, the zero vector will always be in that collection. Is that accurate?

 

[jgens]

Yes. Notice that if $S$ is a set and $v \in S$, then $0 \cdot v = 0 \in \mathrm{Span}(S)$.

 

[QuarkCharmer]

Yes. Notice that if $S$ is a set and $v \in S$, then $0 \cdot v = 0 \in \mathrm{Span}(S)$.

Yup, since the weights of the linear combination can be zero. Thanks, I just wanted clarification due to the books grey area. I know it seems obvious, but you would have to read this book to understand the confusion