If matrix A is row equivalent to matrix B, then row(A) = row(B). This is because the row space of A is just the span of the row vectors of A. The rows of B are a linear combination of the rows of A, so the rows of B lie within the row space of A. And vice versa. Therefore row(A) = row(B)
So why is it that whenever I see a problem asking me to find a basis of the row space of a matrix, the matrix is reduced? Can't you just take the rows of the matrix as they are, and say that those row vectors make up a basis? What's the point of row reducing first?
Side question, whenever you're asked for a basis, would it not be valid to just give the unit vectors of that dimension?
Any help is appreciated.