Let $f(x) = w_1 x_1 + w_2 x_2$, does $\dfrac{\partial f(x)}{\partial x}$ produce a column vector or a row vector?

Question

Suppose that $f(x) = w_1 x_1 + w_2 x_2 = w \cdot x$,

where $w = [w_1, x_2], x = \begin{bmatrix} x_1\\x_2\end{bmatrix}$

Does $\dfrac{\partial f(x)}{\partial x}$

produce a column vector or a row vector?

Answer 1

There's really no difference between a column and a row vector. They're two different ways of writing the same thing.

Furthermore, what's $\partial f/\partial x$?

For a linear map, it can always be represented by a matrix, and the "derivative" turns out to be the same map.