Kernel of Projection in Plane between Lines passing through Origin

Theorem

Let $M$ and $N$ be distinct lines in the plane both of which pass through the origin $O$.

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

$\forall x \in \R^2: \map {\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$.

Then $N$ is the kernel of $\pr_{M, N}$.

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Proof

Let $\LL$ be the straight line through $x$ which is parallel to $N$.

Let $\map {\pr_{M, N} } x = \tuple {0, 0}$.

By definition, $\map {\pr_{M, N} } x$ is the intersection of $M$ with $\LL$.

However, as $\map {\pr_{M, N} } x = \tuple {0, 0}$, it follows that $\LL$ is coincident with $N$.

Hence the result.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.5$