row space
The row space of a matrix is the span of its rows, the set of all vectors you can build by mixing the rows. It lives on the input side of the matrix, in the same space as the vectors x you feed in.
Here is a beautiful fact: the row space and the column space always have the same dimension, and that shared number is the rank. So however lopsided a matrix looks, its independent rows and independent columns agree in count, even though the two spaces usually sit in different places.
The row space also has a precise relationship with the null space: they are orthogonal complements. Every row is perpendicular to every null-space vector, which makes sense because A*x = 0 says each row dotted with x gives zero. Together the row space and null space fill out the entire input space without overlap.
The two rows are the same direction, so the row space is one line; vectors perpendicular to it form the null space.
Row reduction leaves the row space unchanged, which is why the non-zero rows of an echelon form give a clean basis for it.