Ok so I have two quick questions.

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

Find $\displaystyle a$ so that the null space of $\displaystyle A^{T}$ has dimension 0. Find a basis for the column space of $\displaystyle A^{T}$ once $\displaystyle a$ has been determined.

we know A = $\displaystyle \begin{bmatrix} a &1 &-1 &3 \\ 0 &a &-1 &0 \\ 1 &-1 &0 &a \end{bmatrix} $

so I went ahead and set a = 0 so that the dimension of $\displaystyle A^{T}$ = 0.

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

a basis for the column space of $\displaystyle A^{T} $ = all 3 columns of $\displaystyle A^{T}$ which, when reduced, would be =

$\displaystyle SP{\begin{bmatrix} 1\\0\\0\\0 \end{bmatrix} , \begin{bmatrix} 0\\1\\0\\0 \end{bmatrix}, \begin{bmatrix} 0\\0\\1\\0 \end{bmatrix} } $

For my second question, I was wondering if there's a generalized way to find $\displaystyle a$. For this problem, I pretty much did trial and error and looked at what value I could give $\displaystyle a$ so that, when I put $\displaystyle A^{T}$ into R.E.F. it would be linearly independent. Is that the only way to find a value for a?