4. Preservation properties

First, recall that if $T$ is a homogeneous linear transformation then $T$ preserves linear combinations: $T(r$   v$+ s$   w$) = r T($v$) + s T($w$)$. In fact, a transformation $T:$   R$^n \rightarrow$   R$^m$ is homogeneous linear if and only if $T$ preserves all linear combinations.

Affine transformations do not preserve usually linear combinations. For example, if $T$ is a translation, given by $T($x$) =$   x$+$   b with b$\neq$   0, then $T(\frac{1}{2}$   v$+ {\frac{1} {4}}$   w$)$ $=$ $\frac{1}{2}$   v$+ {\frac{1} {4}}$   w$+$   b, but $\frac{1}{2} T($v$) + {\frac{1} {4}} T($w$)$ $=$ $\frac{1}{2} ($v$+$   b$) + {\frac{1} {4}} ($w$+$   b$)$ $=$ $\frac{1}{2}$   v$+ {\frac{1} {4}}$   w$+ {\frac{3} {4}}$   b, not the same thing.

However, there is one kind of linear combination that is preserved:

Theorem 4.1 . Affine transformations preserve linear combinations in which the sum of the coefficients is 1.

Summary of proof. First check the case of a translation. Then combine that case with the case of a homogeneous linear transformation to get the conclusion for all affine transformations.

4.2 . Examples of such linear combinations:

(a) $\frac{1}{2}$   v$+ \frac{1}{2}$   w, the average of two vectors v and w.

(a*) $\frac{1}{2} P + \frac{1}{2} Q$, the midpoint of the line segment joining two points $P$, $Q$. (Remember that points and vectors are essentially the same thing: pairs of numbers, or triples of numbers, etc.)

(b) $(1-t) P + tQ$ for $0 \leq t \leq 1$, the line segment joining $P$ and $Q$.

(b*) $(1-t) P + tQ$ for all $t$, the whole line through $P$ and $Q$.

(c) $P - Q + R$.

Corollary 4.3 . An affine transformation takes line segments to line segments and lines to lines, if it is nonsingular. (If it is singular, it can take a line to a point.)

Actually, Corollary can be improved to an ``if and only if'' statement, i.e., a $\Leftrightarrow$ statement, in the nonsingular case:

Theorem 4.4 . A one-to-one transformation $T:$   R$^n \rightarrow$   R$^n$ takes lines to lines $\Leftrightarrow$ $T$ is affine and nonsingular.

(The proof of the `` $\Rightarrow$'' direction is quite difficult.)