Why does the second vector of a direct sum gets dropped during projection? I encountered this definition in my book on linear algebra "Let a vector space be a direct sum $V = M\oplus N.$ Then for any
$\overrightarrow{v} \in V$ with $\overrightarrow{v} = \overrightarrow{m}+ \overrightarrow{n}$ where $\overrightarrow{m} \in M, \overrightarrow{n} \in N$, the projection of $\overrightarrow{v}$ into M along
N is $Proj_{M,N}(\overrightarrow{v}) = \overrightarrow{m}$." So in the final part of the definition, the $Proj_{M,N}(\overrightarrow{v}) = \overrightarrow{m}$, how does the N not have any effect on the projection? The book gives an example with 2 simple 2 by 2 matrices and some projection A, the N disappears: "The projection of A into M along N keeps the M part and drops the N part."
Thank you!

This is what "projection" means in math. This is the most fundamental purpose of the projection operation: Throw away entirely some information (in this case the $\vec n$-component of $\vec v$) in order to better study what's left (in this case the $\vec m$-component).
