My former multivariable calculus professor denoted the gradient as $$\mathrm{grad}\,\mathit{f}\, (x;y)= \begin{pmatrix}x\\y\end{pmatrix}.$$ Completely forgot that. Since then I've always been writing the gradient as a row vector. Is there a difference? Googled it, but I'm not satisfied. Some even make a difference between the vector derivative and gradient. I'm a bit confused now. What is right? Is there an actual difference?
The reason for my question is a matrix derivate problem. In one of my older question I got a hint to a problem:
$$\frac {\partial(a^Tx)} {\partial x}=\frac {\partial(x^Ta)} {\partial x}=a$$
I tried to learn this rule. My thoughts are as follow. Both numerators are the same because they result in the same scalar. The $x$ in the denominator is a column vector. Therefore the result (here $a$) should be a column vector too when taking the derivate.
On the other hand, taking a vector derivative (or the gradient?) of a scalar (or a function) should result in a row vector $a^T$.
Additionally, if the numerator would be a vector instead of a scalar, then the result would be a Jacobean right? And Jacobeans have on one row all the derivatives respective to a $x_m$, and the $numerators_m$ in the columns. Keeping that form the output of the hint should be $a^T$, shouldn't it?
