# Row equivalent matrices and row spaces

Let $A$ and $R$ be row equivalent $m \times n$ matrices. Let the row vectors of $A$ be $a_1, a_2, a_3, \ldots, a_m$ and the row vectors of $R$ be $r_1, r_2, r_3,\ldots, r_m$. Matrices $A, R$ are row equivalent, therefore the $r$ row vectors are obtained from the $a$ row vectors by elementary row operations. This means that every $r$ row vector is a linear combination of the $a$ row vectors. Therefore the row space of matrix $A$ lies in the row space of matrix $R$.

Is the bolded part in the quote saying $R$ is a subset of row space of $A$? How does that show that row space of $A$ is in the row space of $R$?

It doesn't make any sense to say that "$R$ is a subset of row space of $A$", since $R$ is a matrix and the row space of $A$ is a subspace; they are different objects.
The bolded part implies that every $r$ row vector is in the row space for $A$. That in turn implies that every linear combination of row vectors from $R$ is in the row space for $A$, since the row space of $A$ is a subspace and hence closed under linear combinations. Therefore, the row space of $R$ is in the row space for $A$.
Since we can flip the roles of $R$ and $A$ with no changes to the above, we can also say that the row space of $A$ is in the row space of $R$. Hence, the row space of $R$ and the row space of $A$ are the same.
• The row space of $A$ is a subspace of $\mathbb{R}^n$, if $A$ is $m$ by $n$. It's a subspace because it's closed under vector addition and scalar multiplication. I can provide more details if you need me to. (I would edit my original post, it's too hard to type it out here.) – Nathan H. Jan 3 '17 at 18:26