What Is Projection On Subspace $U$ In Parallel To Subspace $V?$

Question

When I am given that a transformation $T$ is a projection on $U$ in parallel to $V$, the projection "sits" in $V?$

Answer 1

What this means is that $T$ is a linear operator on a vector space $W$, which is the direct sum of $U$ and $V$. That is, for every vector $w \in W$, there is a unique $u \in U$ and $v \in V$ such that $w = u + v$. Further, it means that $T(w) = u$. Since $$T(w) - w = u - (u + v) = -v$$ the line through $w$ and $T(w)$ is always parallel to $V$.

Also $T(w) = 0 \iff w = 0 + v = v \in V$, so as amd said, $V = \ker T$. And $U = \operatorname{im} T$.