Bounds on $b$ when solving $A^{-1}b = x$. I'm very inexperienced with linear algebra, so please bear with me.
For $Ax = b$:
I'm trying to solve for vector, $x$, given $n\times n$ matrix, $A$, and given $b$, where $y\le b\le z$. By bounds on $b$, I mean that the entries of $b$ are filled with parameters, each with their own set of bounds.
So, how do I calculate $x$ with the bounds around $b$? I've read through the Cauchy-Schwarz inequality, and triangle nequality, but still not sure what to do.
Thanks. 
 A: So I understand correctly, you want to characterize
$$\{A^{-1}\mathbf{b}\ \vert\ y_i \leq b_i \leq z_i\};$$
in other words you don't have an exact right-hand side $\mathbf{b}$, but you know that is lies within some high-dimensional rectangular box, and want to know what all possible solutions $\mathbf{x}$ are so that $A\mathbf{x}$ lies within the box.
I'm assuming that $A$ is nonsingular, in which case you know that that since $A^{-1}$ is linear, it maps planes to planes and therefore maps your box to a parallelepiped. You can also easily compute the half-space inequalities that define your parallelepiped: for example let's look at one of your box constraints,
$$A\mathbf{x} \cdot \mathbf{e}_i \geq y_i$$
where $\mathbf{e}_i$ is the Euclidean basis vector. You can rewrite this inequality as a condition on $\mathbf{x}$:
$$\mathbf{x} \cdot A^T\mathbf{e}_i \geq y_i,$$
where of course $A^T\mathbf{e}_i$ is the $i$th row of your matrix. The preimage of your box is thus
$$\{\mathbf{x}\in\mathbb{R}^n\ \vert\ y_i \leq \mathbf{x}\cdot A^T\mathbf{e}_i \leq z_i\}.$$
What you do next, knowing the planes that bound $\mathbf{x}$, will depend on your application.
