Is the gradiant a column or a row?

Question

Suppose we have $f:\mathbb{R^2}\rightarrow \mathbb{R}$. Vectors which $f$ act on are column vectors i.e a $2 \times 1$ matrix.

Is the gradiant $\nabla f$ then a row vector? And why is this logical?

Answer 1

It is a row.

It is logical because the gradient is suppose to be the differential of a function from $\mathbb{R^n}$ to $\mathbb{R^1}$ therefore a linear map and NOT just a vector. In this sense, it is just a $n\times1$ matrix not a vector of $\mathbb{R^n}$

The whole confusion is caused because we can canonically identify the vectors of $\mathbb{R^n}$ with the linear functions from $\mathbb{R^n}$ to $\mathbb{R^1}$ (which is to say that $\mathbb{R^n}^*\cong \mathbb{R^n}$).