I hope this one is easy. Suppose I have an underdetermined, rectangular matrix $A$ and vector $b$. I want to reason about the subspace where $Ax = b$ and specifically the projection $y:= Tx$. Is there a way to describe the space of $y$'s that satisfy the constraint $Ax = b$? My intuition is this should also be a linearly constrained subspace.
I.e., how do I express constraints that define the space $\{y | \exists x, Ax = b, y = Tx \}$?