Lemma 1
If tεA(2) and s1,s2...sn are real numbers with
s1+s2+...+sn = 1, then
for any vectors p1,...,pn,
t(Σsipi) = Σsit(pi).
Proof
Let p = Σsipi, and let t(x) = Ax+b. Then
| t(p) |
= Ap+b |
|
|
= A(Σsipi) + Σsib, |
as Σsi = 1 |
|
= Σsi(Api+b) |
|
|
= Σsit(pi) |
|
Lemma 2
If tεA(2) fixes points P and Q, then it fixes every point of the line PQ.
Proof
Let P and Q have position vectors p and q respectively. Then any R on PQ
has position vector r = sp+(1-s)q, for some real s. Since s+(1-s) = 1, we
can apply Lemma 1 to get t(r) = st(p)+(1-s)t(q) = sp+(1-s)q = r,
since t
fixes P and Q. Thus t fixes R.
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