# Why are the row spaces of equivalent matrices equal?

If you have matrixes $A$ and $B$, and $A \sim B$, $\text{row} (A) = \text{row} (B)$. I noticed this pattern in tons of matrixes and was wondering why it was true. Note the tilde sign means the matrices are equivalent by $B$ has had some elementary row operations performed on it. (i.e. B is the row echelon form of A)

• Perhaps it is true because no elementary row operation changes the row space? Oct 27 '13 at 17:41

Each row in $$B$$ is a linear combination of the rows in $$A$$, hence $$\operatorname{row}(B)\subseteq \operatorname{row}(A)$$. Since the steps can be reverted, we also have $$\operatorname{row}(B)\supseteq \operatorname{row}(A)$$.
• Could you explain why the rows in B being a linear combination of the rows in A imply the row space of A $\subseteq$ the row space of B? Oct 27 '13 at 17:47