Vector Space vs. Span I have hard time understanding why span of some vectors is not exactly their vector space but rather the smallest subspace
 A: As an example consider $\mathbb{R}^2$, and then you have that $\text{Span}\begin{bmatrix}1\\0\end{bmatrix}$ will just consist of vectors of the form $\begin{bmatrix}x\\0\end{bmatrix}$. Thus, the span is just the $x$-axis which is a subspace of the vector space $\mathbb{R}^2$.
A: Not to over contribute, but I think it's smart to hammer home that as lulu mentioned, without defining their vector space the statement really has no meaning. You can't talk about a subset not spanning the space without first saying what the space is.
Like Steven Creech mentioned, the span of $\{(1,0)\}$ in $\mathbb{R}^2$ is just the x-axis, but the span of $\{(1,0),(0,1)\}$ is $\mathbb{R}^2$, which is the vector space they live in (as you mentioned in your question).
Lastly just to show you why the span is the smallest subspace, if you have some arbitrary $S \subseteq V$ where $V$ is a vector space over $\mathbb{F}$, then the span of $S$ is the smallest subspace which contains S. To see this, note that if a random subspace is to contain $S$, because it is closed under vector addition and scalar multiplication; it must also contain all linear combinations of the elements of $S$, in other words it has to contain the span of $S$. So the smallest subspace that contains $S$ is it's span.
