Another linear operation that you are familiar with is anti-differentiation, or indefinite integration. In this problem, we will study features of antidifferentiation and relate it to differentiation.

(a) First, demonstrate that anti-differentiation is a linear operator. For antidifferentiation to be linear, what value must any integration constant be set to?

(b) If anti-differentiation is a linear operator, then we expect it in general to be able to be expressed as a matrix. Let's call this matrix \(\mathbb{A}\) and assume that it acts on function vectors defined on a spatial grid. For a derivative matrix \(\mathbb{D}\) and function vectors \(\vec{f}, \vec{g}\), we assume \[\begin{equation*}\mathbb{D}\vec{f}=\vec{g} \tag{2.93}\end{equation*}\]We can solve for \(\vec{f}\) given \(\vec{g}\) as\[\begin{equation*}\vec{f}=\mathbb{D}^{-1}\vec{g}=\mathbb{A} \vec{g}, \tag{2.94}\end{equation*}\] because anti-differentiation is the inverse operation of integration. However, does this exist, in general? If it does not exist, what additional constraint must you impose for the anti-differentiation matrix \(\mathbb{A}\) to be well-defined?