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 $\operatorname{pr}_{M, N}$ be the projection on $M$ along $N$.

$M$ and $N$ are respectively the codomain and kernel of $\operatorname{pr}_{M, N}$.


 * $\operatorname{pr}_{M, N} \left({x}\right) = x \iff x \in M$

If $M$ is the $x$-axis and $N$ is the $y$-axis, then $\operatorname{pr}_{M, N} \left({\lambda_1, \lambda_2}\right) = \left({\lambda_1, 0}\right)$.

If $M$ is the $y$-axis and $N$ is the $x$-axis, then $\operatorname{pr}_{M, N} \left({\lambda_1, \lambda_2}\right) = \left({0, \lambda_2}\right)$.

Any such projection is a linear operator.