Dimension is the number of vectors in a basis, not the number of vectors displayed.
WHAT HAPPENS IF…Can three vectors span a plane rather than all of space?
Interactive exploration
Loading the mathematical model…
Connect the picture to the ideadimspan{e1,e2,(1,1,t)}={23t=0t=0
The first two standard vectors span the xy-plane. When t=0 the third is their sum and adds no new direction. For nonzero t the determinant is t, giving a third independent direction. With no generators, the span is {0} and has dimension zero.
A detail that matters
An affine plane away from the origin is not a vector subspace, although it has affine dimension two.