0
$\begingroup$

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 $R$?

$\endgroup$
0
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ @ Nathan H., row space of A is a subspace of what or how do we know it's a subspace? Thanks. $\endgroup$ – Chimp Jan 3 '17 at 18:22
  • $\begingroup$ 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.) $\endgroup$ – Nathan H. Jan 3 '17 at 18:26
  • $\begingroup$ @ Nathan H., I see what's up. Cheers. $\endgroup$ – Chimp Jan 3 '17 at 18:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.