Projection in Plane between Lines passing through Origin is Linear Operator

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

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

$$M$$ and $$N$$ are respectively the range and kernel of $$p_{M, N}$$.

$$p_{M, N} \left({x}\right) = x \iff x \in M$$.

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

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

Any such projection is a linear operator.