Understanding how to find a basis for the row space/column space of some matrix A.

Understanding how to find a basis for the row space/column space of some
matrix A.

I just need some verification on finding the basis for column spaces and
row spaces.
If I'm given a matrix A and asked to find a basis for the row space, is
the following method correct?
-Reduce to row echelon form. The rows with leading 1's will be the basis
vectors for the row space.
When looking for the basis of the column space (given some matrix A), is
the following method correct?
-Reduce to row echelon form. The columns with leading 1's corresponding to
the original matrix A will be the basis vectors for the column space.
When looking for bases of row space/column space, there's no need in
taking a transpose of the original matrix, right? I just reduce to row
echelon and use the reduced matrix to get my basis vectors for the row
space, and use the original matrix to correspond my reduced form columns
with leading 1's to get the basis for my column space.