I'm given the following augmented matrix and asked to find a solution using a generalized inverse: $$ \left[\begin{array}{ccc|c} 2 & 0 & -2 & 2\\ 0 & 2 & -2 & -2\\ -2 & -2 & 4 & 0\\ \end{array}\right] $$ I found the following solution and general solution form: $$ \left[\begin{array}{c} 1 \\ -1\\ 0\\ \end{array}\right];\left[\begin{array}{c} 1+a \\ -1+b\\ 0+c\\ \end{array}\right]$$
Where a,b,c can be any value. I know that given my solution vector, any other solution where a=b is a linear combination of that vector and therefore not a linearly independent solution, but I'm not sure how to proceed when asked to find the set of all linearly independent solutions, especially since my current understanding is that there are infinitely many.
First of all, there is only one free variable $c$ and the general solution is
$$ \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} + c \cdot \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}. $$
Secondly, no two solutions to a nonhomogeneous system are linearly dependent. Thus, the set of all solutions IS the set of all linearly independent solutions.