Given A has the same row space as rref A, do they both have the same basis in general

Question

In Strang’s book in the section on the Dimensions of the Four Subspaces, he says, “ A has the same row space as R. Same dimension r and same basis.“

i believe they have the same dimension and same space, but different bases. In other word I believe A and R have different bases in general, but a basis of each spans the row space. In other words I think it’s wrong to say A and R have the same basis. Do you think it’s wrong to say they have the “same basis”??

Answer 1

It should be clear that the row space of matrix A is the same as the row space of rref(A). This is because the row space of A is invariant under elementary operations (linear combinations). Now, remember that a basis is a smallest set of linearly-independent vectors which span the space of interest. It is certainly not unique.

So, if B is a basis for row(A), it is also a basis for row(rref(A)). Likewise, if B' is a basis for row(rref(A)), then it is also a basis for row(A). Because the rref operations are invertible, you should always be able to move from one basis to another. If you find different basis vectors for row(A) and row(rref(A)), don't worry, you should be able to go from one basis to another. Therefore, the two spaces also have the same basis.