Question

Are column transformations valid to do while finding solution to Ax = 0. Doesn't it modify the nullspace? In what situations is it okay to do column transformations?

Answer 1

Column transformations are never valid when trying to find the null space. If you think about the matrix as a series of linear equations, a column transformation would change the coefficients of one variable in each equation, without changing the solution to that equation. The result is not the same equation you started with.

Answer 2

When you perform column operations, you have to account for any transformation you are performing, because, as kutulu remarked, you are generally altering the solution:

Ex: x+y=0 => x=-y
transform second column e.g. scale by 2:
x+2y=0 => y'=2y
now: x=-2y' => x=-2(1/2)*y=-y

Interchange columns one and two:
x'+y'=0, x'=y y'=x

Similarly, you'd have to account for replacing column k by (column k) + (a*column j)