Projection of Straight Line on Another in Plane
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 $\operatorname{pr}_{M, N}$.
As the kernel is a concept defined in relation to a homomorphism, it needs to be clarified what homomorphism is being considered.
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- $\operatorname{pr}_{M, N} \left({x}\right) = x \iff x \in M$
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.
Proof
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Sources
- Seth Warner: Modern Algebra (1965): $\S 28$: Example $28.5$
