Why is it not possible for the columns of a matrix to span R^2 even if

[fackert]

Why is it not possible for the columns of a matrix to span R^2 even if those columns do not span R^3?

 

[Bacle2]

You may end up with a rank-1 matrix after row-reduction.

 

[HallsofIvy]

Why is it not possible for the columns of a matrix to span R^2 even if those columns do not span R^3?

Is that really what you meant to ask? There is no "why" for a statement that isn't true to begin with! The columns of the matrix
$$\begin{bmatrix}1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & 0\end{bmatrix}$$
do span R² but not R³.

(Bacle answered a different question: Why is it possible that the columns of a matrix do not span R²?)