It seems like $a^Tb$ is the same as writing $a \cdot b$ in matrix form.
1) Why is $n \times 1$ and $n \times 1$ matrix multiplication undefined?
2) Is this just a generalization of the dot product?
If so, is it just a way to generalization the orthogonal projection of one vector onto another, as a way to find the closest point on the second vector to the first vector? Does:
$$\frac{a\cdot b}{a\cdot a} = \frac{a^Tb}{a^Ta}$$?
And then, that quantity should equal p, which if I'm not mistaken equals yhat?