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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)

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  • $\begingroup$ Perhaps it is true because no elementary row operation changes the row space? $\endgroup$
    – Carsten S
    Oct 27 '13 at 17:41
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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 $\operatorname{row}(B)\supseteq \operatorname{row}(A)$.

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  • $\begingroup$ 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? $\endgroup$
    – user103462
    Oct 27 '13 at 17:47

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