Abstract: Let $A$ be a subvariety of affine space $\mathbb{A}^n$ whose irreducible components are $d$-dimensional linear or affine subspaces of $\mathbb{A}^n$. Denote by $D(A)\subset\mathbb{N}^n$ the set of exponents of standard monomials of $A$. Using the Hilbert function, we show that $D(A)$ contains as many subspaces of dimension $d$ as $A$ contains irreducible components. We refine this result in various ways. Firstly, we specify the directions into which the subspaces of highest dimension of $D(A)$ point. Secondly, we identify $A$ as the family of $A_{\lambda}=A\cap{X_{1}=\lambda}$, where $\lambda$ runs through $\mathbb{A}^1$. In the the case where all components of $A$ are contained in some $A_{\lambda}$, we give a complete description of $D(A)$ in terms of the $D(A_{\lambda})$. In complementary case, we find an open $U\subset\mathbb{A}^1$ such that $D(A_{\lambda})$ is constant on $U$, and trace both the generic and the nongeneric $D(A_{\lambda})$ in $D(A)$. We use the moduli space of all $A$ as above, and a Gröbner cover of this space.