Dimension of the span

Question

So i have this question :

The number of columns of a matrix gives the dimension of the span of the column vectors.

I know that the answer is indeed FALSE.

But i want to back my answer up with a simple, right, yet CONCRETE explanation of why that is so.

I was thinking : the question if false due to the fact that the dimension o of the span of the column vectors is given by the maximum amount of linearly independent vectors.

is that correct?

Answer 1

Take then null $n\times n$ matrix. It has $n$ columns, but the dimension of the span of the column vectors is $0$.