Definition: The rank of a matrix A is the dimension of the row space of A.
I have seen this definition in a few other posts as well, but I am wondering why the definition isn't more clear like this:
The rank of a matrix A is the dimension of the basis for either the row/col space of A.
The original definition seems a little too confusing so I am wondering if I have misunderstood anything because I know there is a difference between row space and the basis for a row space?
On
"Dimension of a basis for..." : this is the first problem. Bases don't have a dimension. Vector spaces/subspaces have a dimension. The row space is a subspace, a basis for it is not.
"...for either the row/col space..." : this is a second problem. As written, it makes this proposed definition ambiguous. Which is it? The row space or the column space? Before you answer with "they both have the same dimension", you should realize that that is a further result that needs to be proven, and shouldn't be involved in the definition.
The common definition is very clear and, whether or not you find it confusing, unambiguous. The row space of a matrix is something that is well defined, and has a dimension. We call that dimension the rank of the matrix.
Dimension is the number of elements in a basis of a vector space. A basis is an independent set that span the space.
We define the rank to be dimension of the row space. We can't define it to be dimension of row space or column space without justification first. We later verify that the dimension of row space is equal to the dimension of column space, this is done by a proof rather than a definition.