Projection in Plane between Lines passing through Origin is Linear Operator

Theorem
Let $M$ and $N$ be distinct straight lines through the plane through the origin.

Let $\pr_{M, N}$ be the projection on $M$ along $N$.

$M$ and $N$ are respectively the codomain and kernel of $\pr_{M, N}$.


 * $\map {\pr_{M, N} } x = x \iff x \in M$

If $M$ is the $x$-axis and $N$ is the $y$-axis, then $\map {\pr_{M, N} } {\lambda_1, \lambda_2} = \tuple {\lambda_1, 0}$.

If $M$ is the $y$-axis and $N$ is the $x$-axis, then $\map {\pr_{M, N} } {\lambda_1, \lambda_2} = \tuple {0, \lambda_2}$.

Any such projection is a linear operator.