Why don't the dimensions in this linear regression equation match up?
My guess is that $x$ here refers to a single datapoint with shape $m$x$1$. So $h_w(x)$ and $w^Tx$ both have shape $1$. In the case of multiple datapoints given as a matrix $X$ of shape $n$x$m$, the equation becomes:

$h_w(X) = Xw$