Is Wx+b a hyperplane? I know $w^T x = b$ is. The intuition here is clear with w being a normal vector perpendicular to the various $x$ vectors making up the translated hyperplane.
However, when you have $Wx = b$, where $W$ is a matrix, $x$ is a vector, and $b$ is a vector, I don't understand the geometry. It's like each row of $w_1^T x = b_1$ is a hyperplane, but what is the geometry of the whole expression?
 A: Suppose $x \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$ so $W$ is an $m \times n$ matrix.
One way to think about the equation $Wx = b$ is as a system of $m$ equations--one for each element of $b$. Writing them out (and letting $w_{(i)}$ denote the $i$th row of $W$), we have:
$$\begin{array}{c}
    w_{(1)} \cdot x = b_1 \\
    \vdots \\
    w_{(m)} \cdot x = b_m \\
\end{array}$$
The solution to $Wx=b$ is the set of all vectors that satisfy all of these equations.
We can think of the solution to each equation as a geometric object. As you noticed, this is a hyperplane (with the exceptions that, when $w_{(i)} = \vec{0}$, the solution is the entire space $\mathbb{R}^n$ if $b_i = 0$ and doesn't exist if $b_i \ne 0$). From a geometric perspective, the solution to $Wx=b$ is the intersection of all of these objects. It may be a hyperplane in certain cases, but need not be in general.
For example, let's consider the easily visualizable case where $n=3$, and solutions to the individual equations are regular 2d planes. If there are $m=2$ planes, they can intersect in a line or completely overlap. Or, if they're parallel, they don't intersect and no solution exists. If there are $m=3$ or more planes, they might all intersect at a point or in a line. Or, they might completely overlap or fail to intersect.
