Understanding the difference between Span and Basis I've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for.
I understand that the Span of a Vector Space $V$ is the linear combination of all the vectors in $V$.
I also understand that the Basis of a Vector Space V is a set of vectors ${v_{1}, v_{2}, ..., v_{n}}$ which is linearly independent and whose span is all of $V$.
Now, from my understanding the basis is a combination of vectors which are linearly independent, for example, $(1,0)$ and $(0,1)$. 

But why?

The other question I have is, what do they mean by "whose span is all of $V$" ?
On a final note, I would really appreciate a good definition of Span and Basis along with a concrete example of each which will really help to reinforce my understanding.
Thanks.

The span of a finite subset $S$ of a vector space $V$  is the smallest subvector space that contains all vectors in $S$. One shows easily it is the set of all linear combinations of lelements of $S$ with coefficients in the base field (usually $\mathbf R,\mathbf C$ or a finite field).
A basis of the vector space $V$ is a subset of linearly independent vectors that span the whole of $V$. If $S=\{x_1, \dots, x_n\}$ this means that for any vector $u\in V$, there exists a unique system of coefficients such that 
$$u=\lambda_1 x_1+\dots+\lambda_n x_n. $$
