I have a question about the following excerpt from p.93-94 in a paper of Donaldson:
Suppose $U_0, U_1$ are finite-dimensional vector spaces and $\Gamma$ is a linear subspace of $U_0 \oplus U_1$. Then, up to a scalar factor, $\Gamma$ defines a Plucker point $|\Gamma|$ in the exterior algebra $\Lambda^*\left(U_0 \oplus U_1\right)$. In turn, up to a scalar ambiguity, elements of this exterior algebra can be viewed as linear maps from $\Lambda^*\left(U_0\right)$ to $\Lambda^*\left(U_1\right)$, so we have $$ |\Gamma|: \Lambda^*\left(U_0\right) \rightarrow \Lambda^*\left(U_1\right) $$ defined up to a scalar ... In general the construction satisfies a composition rule, for the case when one has another subspace $\Gamma^{\prime} \subset U_1 \oplus U_2$, provided the subspaces satisfy a transversality condition. One wants the sum of the projection maps from $\Gamma \oplus \Gamma^{\prime}$ to $U_1$ to be surjective. Then if one defines $\Gamma^{\prime \prime} \subset U_0 \oplus U_2$ to be set of pairs $\left(u_0, u_2\right)$ for which there exists a $u_1 \in U_1$ with $\left(u_0, u_1\right) \in \Gamma,\left(u_1, u_2\right) \in \Gamma^{\prime}$, one has $\left|\Gamma^{\prime \prime}\right|=\left|\Gamma^{\prime}\right| \circ|\Gamma|$, if one uses an appropriate rule for normalising the scalar ambiguities.
My question is in regards to "One wants the sum of the projection maps from $\Gamma \oplus \Gamma^{\prime}$ to $U_1$ to be surjective." What is the sum of the projection maps explicitly? Why do we want it to be surjective for this composition to work?