Ok so I have two quick questions.

The problem I'm trying to solve is as follows:

Find so that the null space of has dimension 0. Find a basis for the column space of once has been determined.

we know A =

so I went ahead and set a = 0 so that the dimension of = 0.

With that, I was wondering if I would instantly be able to say:

a basis for the column space of = all 3 columns of which, when reduced, would be =

For my second question, I was wondering if there's a generalized way to find . For this problem, I pretty much did trial and error and looked at what value I could give so that, when I put into R.E.F. it would be linearly independent. Is that the only way to find a value for a?