When do two matrices have the same column space? Recently I started learning about matrices and know for example that the pivot columns of a matrix form a basis for the column space of this matrix.
I just can't seem to find out when two matrices have the same column space. Wouldn't it be enough to show that both the matrices have the same reduced echolon form (like with the null space) and therefore the same basis for the column space and thus the same column space?
Can someone help me on this?
Edit:
Maybe another question: how can you tell that two matrices don't have the same column space? Can I for example use the fact that if two matrices don't have the same dimension for the column space, that the column spaces cannot be equal?
 A: When you row-reduce a matrix, the dimension of the column space stays fixed, so if $A,B$ have the same reduced echolon form then the dimensions of the column spaces are equal, but the column spaces might not be equal: 
$$A=\begin{pmatrix}1&2\\1&2\end{pmatrix}\hspace{10pt}B=\begin{pmatrix}1&2\\2&4\end{pmatrix}$$
The have the same reduced echolon form, but different column-spaces.
In general, the only way to make sure that two matrices have the same column space is to column-reduce them (unless both are of full rank).
A: Let $A, B \in \mathrm{M}_{m,n}(\mathbb{K})$ be two matrices with $m$ rows, $n$ columns and entries in the field $\mathbb{K}$.
The following statements are equivalent:


*

*there exists an invertible matrix $P \in \mathrm{GL}_m(\mathbb{K})$ such that $B = P A$;

*there exists a finite sequence of elementary Gauss operations on the rows that transforms $A$ into $B$;

*the subspaces of $\mathbb{K}^m$ generated by the rows of $A$ and $B$, respectively, are equal.


The following statements are equivalent:


*

*there exists an invertible matrix $P \in \mathrm{GL}_n(\mathbb{K})$ such that $B = A P$;

*there exists a finite sequence of elementary Gauss operations on the columns that transforms $A$ into $B$;

*the subspaces of $\mathbb{K}^n$ generated by the columns of $A$ and $B$, respectively, are equal.
