It is obvious that elementary row operations will not change the dimension of the row space of a matrix. But is there an easy way to understand that it also will not change the dimension of the column space? For example, I can understand that swapping two rows has no effect, because this is just to switch the order of two basis vectors in the column space. What about the other two operations?
All you need to see is that elementary row operations do not change linear dependence relations among the columns; pick a set of columns, they will be linearly dependent after the row operation if and only if they were dependent before the row operation. And the way to see this is to note that when you multiply the matrix by a vector, you get a linear combination of the columns of the matrix, the coefficients in the linear combination being the entries of the vector. So a linear dependence among the columns corresponds to an element of the nullspace, and row operations don't affect the nullspace (that's the idea behind using row operations to solve a system of equations).