Let $V$ be an n-dimensional real vector space decomposed as a direct sum: $V=P\oplus Q$, with $\dim P=k$ and $\dim Q=n-k$. The graph of any linear map $X:P\rightarrow Q$ can be identified with a k-dimensional subspace $\Gamma(X)=\{v+Xv : v\in P\}$. Any such subspace has the property that its intersection with Q is the zero subspace.
I don't understand the last sentence. How can $\Gamma(X)$ not intersect $Q$ if its elements are a linear superposition of elements from $v\in P$ and $Xv\in Q$?