By Gottfried Barthel, Friedrich Hirzebruch, Thomas Höfer

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**Example text**

Then y = λp + (1 − λ)q for some λ with 0 ≤ λ ≤ 1. Let us determine f (y): f (y) = b, y − c = b, λp + (1 − λ)q − c = λ b, p + (1 − λ) b, q − c = λ( b, p − c) + (1 − λ)( b, q − c) = λf (p) + (1 − λ)f (q) . 46 2 TRANSFORMATIONS If, for example, p ∈ Hl1 and q ∈ Hl1 , then f (p) > 0 and f (q) > 0. It then follows from the computation that f (y) > 0, whence y ∈ Hl1 . If p ∈ Hl1 and q ∈ Hl2 , then f (p) > 0 and f (q) < 0. We can easily check that in this case λ0 p + (1 − λ0 )q ∈ l for λ0 = −f (q)/(f (p) − f (q)) .

If a = o, the line is unique. The line l is then called the span of a. We can easily verify that λa + μa = (λ + μ)a. Let us now consider addition: if a and b have distinct spans, then o, b, a + b, x2 x2 l a+b a 2a a o −a b o x1 x1 Fig. 23. (a) Scalar multiplication; (b) addition and a are, in this order, the vertices of a parallelogram; we say that this parallelogram is spanned by the vectors a and b. To prove this it suﬃces to show that the line l through o and b is parallel to the line m through a and a + b.

The name reﬂection is evocative. If we imagine a physical mirror placed perpendicularly to this page, see Fig. 2, the eye sees both the ﬁgure F and mirror l eye Fig. 2. Construction of the reﬂected rays using the mirror image its reﬂection Sl (F ). In fact, we use the virtual extension inside the mirror of the light rays through which we see the reﬂected image to reconstruct the real light rays. The reﬂection axis is also called an axis of rotation. The image that is associated to this name is the plane turning over in space, where the reﬂection axis is used as rotation axis.