Column Space vs Span and minimum size of a set of vectors to guarantee a vector is in span?

The notation $operatorname{span}(X)$ should be used for a set $Xsubseteq V$ that is contained in some vector space $V$. It is then defined as the intersection of all subspaces of $V$ containing $X$ or equivalently as the set of all linear combinations of elements from $X$.

Writing "$operatorname{span}(A)$" when $A$ is a matrix, doesn't make any sense in terms of this definition. However, it might be an abuse of notation to denote the span of the set of columns of $A$, that is, the column space of $A$. This space is usually denoted $operatorname{col}(A)$ though.

The question about a "minimum number of vectors in a set $X$ to guarantee $binoperatorname{span}(X)$" is unclear to me. For example in the plane $V=mathbb R^2$ you can take the infinite set $$ X = left{ begin{bmatrix} 1 \ 0 end{bmatrix}, begin{bmatrix} 2 \ 0 end{bmatrix}, begin{bmatrix} 3 \ 0 end{bmatrix}, begin{bmatrix} 4 \ 0 end{bmatrix}, dots right} $$ with span $$ operatorname{span}(X) = left{ begin{bmatrix} x \ 0 end{bmatrix} ,middle|, xinmathbb R,right} $$ the does not contain $b=begin{bmatrix}0 \ 1end{bmatrix}$ despite $X$ having infinitely many elements.

For example

Let A= begin{bmatrix} 1 &0 &0&0\ 0& 1 &0&1\ 0 &0& 1&1\ end{bmatrix}.

Here $Col(A)= mathbb{R}^3$ and observe that last column in A is sum of 2nd and 3rd columns.

Note that $Span(A)=Col(A)=Row(A)$