# linear function

Let $\mathscr{S}_{1}=(\mathcal{P}_{1},\mathcal{L}_{1})$ and $\mathscr{S}_{2}=(\mathcal{P}_{2},\mathcal{L}_{2})$ be two near-linear spaces.

Definition. A linear function from $\mathscr{S}_{1}$ to $\mathscr{S}_{2}$ is a mapping on the points that sends lines of $\mathscr{S}_{1}$ to lines of $\mathscr{S}_{2}$. In other words, a linear function is a function $\sigma:\mathcal{P}_{1}\to\mathcal{P}_{2}$ such that

 $\sigma(\ell)\in\mathcal{L}_{2}\mbox{ for every }\ell\in\mathcal{L}_{1}.$

Here, $\sigma(\ell)$ is the set $\{\sigma(P)\mid P\in\ell\}$. A linear function is also called a homomorphism.

When both $\mathscr{S}_{1}$ and $\mathscr{S}_{2}$ are linear spaces, then $\sigma$ being a linear function is equvalent to saying that $P,Q$ are collinear iff $\sigma(P),\sigma(Q)$ are collinear.

If $\mathscr{S}_{1}$ is a linear space, then so is $(\sigma(\mathcal{P}_{1}),\sigma(\mathcal{L}_{1}))$. This shows that if $\sigma:\mathscr{S}_{1}\to\mathscr{S}_{1}$ is onto, $\mathscr{S}_{2}$ is a linear space if $\mathscr{S}_{1}$ is.

Let $\sigma:\mathscr{S}_{1}\to\mathscr{S}_{2}$ be a one-to-one linear function. If points $P_{1}\neq P_{2}$ lie on line $\ell$, then $\sigma(P_{1})\neq\sigma(P_{2})$ lie on $\sigma(\ell)$. This also shows that three collinear points in $\mathscr{S}_{1}$ are mapped to three collinear points in $\mathscr{S}_{2}$. In addition, we have

$|\ell|=|\sigma(\ell)|$ for any line $\ell$ in $\mathscr{S}_{1}$.

Definition. When $\sigma:\mathscr{S}_{1}\to\mathscr{S}_{2}$ is a bijection whose inverse $\sigma^{-1}$ is also linear, we say that $\sigma$ is an isomorphism. When $\mathscr{S}_{1}=\mathscr{S}_{2}=\mathscr{S}$, we call $\sigma$ an automorphism, or more commonly among geometers, a collineation, of the space $\mathscr{S}$.

Suppose $\sigma:\mathscr{S}_{1}\to\mathscr{S}_{2}$ is an isomorphism. For every point $P$, let $P^{*}$ be the set of all lines passing through $P$. Then

$|P^{*}|=|\sigma(P)^{*}|$ for any point $P$ in $\mathscr{S}_{1}$.

It is possible to have a bijective linear function whose inverse is not linear. For example, let $\mathscr{S}_{1}$ be the space with two points $P,Q$ with no lines, and $\mathscr{S}_{2}$ the space with the same two points with line $\{P,Q\}$. Then the identity function on $\{P,Q\}$ is a bijective linear function whose inverse is not linear. On the other hand, if the both spaces are linear, then the inverse is always linear.

Remark. The usage of the term โlinear functionโ differs from its more usual meaning as a linear transformation between vector spaces in the study of linear algebra.

## References

• 1 L. M. Batten, , 2nd edition, Cambridge University Press (1997)
Title linear function LinearFunction 2013-03-22 19:14:46 2013-03-22 19:14:46 CWoo (3771) CWoo (3771) 12 CWoo (3771) Definition msc 51A45 msc 51A05 msc 05C65