  The space of smooth sections of an equivariant line bundle over the real
projective space $\mathbb{R}{\rm P}^n$ forms a natural representation of the
group ${\rm GL}(n+1,\mathbb{R})$. We explicitly construct and classify all
intertwining operators between such representations of ${\rm
GL}(n+1,\mathbb{R})$ and its subgroup ${\rm GL}(n,\mathbb{R})$, intertwining
for the subgroup. Intertwining operators of this form are called symmetry
breaking operators, and they describe the occurrence of a representation of
${\rm GL}(n,\mathbb{R})$ inside the restriction of a representation of ${\rm
GL}(n+1,\mathbb{R})$. In this way, our results contribute to the study of
branching problems for the real reductive pair $({\rm GL}(n+1,\mathbb{R}),{\rm
GL}(n,\mathbb{R}))$. The analogous classification is carried out for
intertwining operators between algebraic sections of line bundles, where the
Lie group action of ${\rm GL}(n,\mathbb{R})$ is replaced by the action of its
Lie algebra $\mathfrak{gl}(n,\mathbb{R})$, and it turns out that all
intertwining operators arise as restrictions of operators between smooth
sections.
