Is the row space of a matrix A a subspace A? If so, what are the objects of A?

Question

I have problems with an assertion that I read in a definition of the row space. I hope somebody can help me :)

This part is clear: Let A be a mxn-matrix, with rows $ r_{1},...,r_{m} \in K^{n} $ The set of all possible linear combinations of $ r_{1},...,r_{m} $ is the row space of A. Also the row space is a subspace of $K^{n}$.

My Problem:

In one definition that I have read it is stated that the row space of A is also a subspace of A. So the concrete Matrix A has to be a vector space. What are the objects of A?

Can somebody give me an interpretation of what it exactly means that the row space of A is a subspace of A?

PS: I'm not used to write about math in english, please ask if something doesn't makes sense to you.

Answer 1

A matrix is not a vector space. Generally, it represents a linear transformation, from one vector space to another.

Now, since the number of columns always equals the dimension of the domain space, we can view the row space as a subspace of $V$ (where $T:V\to W$).

Answer 2

No, a matrix is not a vector space. Asserting that the row space of a matrix is a subspace of that matrix is just an abuse of language.