The Set $\{ Ax + b | Fx = g \}$, is it affine? How can I prove it?
My answer is yes, the intuition is that $\{ x | Fx = g \}$ is a solution space of equation $Fx = g$, thus it is a linear subspace.
The $Ax + b$ is a linear transformation plus a translation. So the final result is a affine set.
Welcome to comment on my understanding.
Hint:
1. A set is an affine subspace, iff it is the kernel of an affine transformation $T: x \mapsto Mx+c$
2. An image of an affine subspace under an affine transformation is an affine subspace.