Understanding the difference between Span and Basis

Span is usually used for a set of vectors. The span of a set of vectors is the set of all linear combinations of these vectors.

So the span of $\{\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ would be the set of all linear combinations of them, which is $\mathbb{R}^2$. The span of $\{\begin{pmatrix}2\\0\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ is also $\mathbb{R}^2$, although we don't need $\begin{pmatrix}2\\0\end{pmatrix}$ to be so.

So both these two sets are said to be the spanning sets of $\mathbb{R}^2$.

However, only the first set $\{\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ is a basis of $\mathbb{R}^2$, because the $\begin{pmatrix}2\\0\end{pmatrix}$ makes the second set linearly dependent.

Also, the set $\{\begin{pmatrix}2\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}\}$ can also be a basis for $\mathbb{R}^2$. Because its span is also $\mathbb{R}^2$ and it is linearly independent.

For another example, the span of the set $\{\begin{pmatrix}1\\1\end{pmatrix}\}$ is the set of all vectors in the form of $\begin{pmatrix}a\\a\end{pmatrix}$.