Let's say we have a gradient vector:
$$\nabla (x) \in R^d$$
Since the gradient is also a function (just like how in 1D, the derivative of a function is also a function), then we can say:
$$\nabla: R^d\longrightarrow R^d$$
That is, the gradient transforms inputs of $R^d$ to yield the gradient at that point.
Can we represent the gradient as a matrix $R^d \times R^d $ in this case? (Since the gradient is a linear operator). Is it useful/used in practice?
Note: A follow-up is that the gradient "matrix" may not be square, since we may need an extra term to represent affine transforms. What about this case?