A,B matrices and column space

Question

if A,B have the same column space , does that means they have the same number of rows and the same rank?

Answer 1

It does mean the matrices have the same number of rows since otherwise one would not be able to compare the column spaces. E.g.

$$A = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, B = \begin{pmatrix} 1 \\ 0\end{pmatrix}$$

Furthermore it also means that the rank is the same. For this look at why the row rank equals the column rank (and the column rank just being the dimension of the column space)