Vector Span, Linear Independence, Column Space, and Null Space
The span of vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$ is the set of every linear combination $c_1 \mathbf{v}_1 + \cdots + c_k \mathbf{v}_k$. The shaded region below is the span. The black arrow is the linear combination chosen by the sliders, drawn tip to tail with dashed arrows.
Top, front, and side views
A basis for the span
The span of the vectors is the column space of the matrix $A$ whose columns are the vectors. Row reducing $A$ shows which columns are needed.