Are there matrices with same column space but different rank

Question

Is there exist some matrices $A,B$ that they have the same column space but different rank? (I do not get if this matrices is $n\times n$ or $m\times n$).

I think that they share the same column space something like this

A=$\begin{bmatrix} 1& 0\\ 0& 0 \end{bmatrix}$ B=$\begin{bmatrix} 1& 0\\ 0& 1 \end{bmatrix}$.

Here they have the same one vector from column space, but I do not know is this meaning that they need to have every vector of column space the same or not, what do you mean?

Answer 1

The rank of a matrix is equal to the dimension of the column space (and also equal to the dimension of the row space). Therefore, the answer is negative.