# Math 22a Fall 2018

## 22a Linear Algebra and Vector Analysis

# Pythagoras

Pythagoras of Samos lived from 570 to 495 BC. The Pythagorean theorem applies in any vector space with inner product assigning to two vectors a number. The theorem is the relation a^{2}+b

^{2}=c

^{2}, where a=|v|, b=|w| and c=|v-w| are lengths of a right angle triangle formed by the points A, B=A+v, C=A+w where AB=v, AC=w, CB=v-w are the vectors connecting the points. The right angle assumption is written mathematically in the form v.w=0. The length is defined as |v| = (v.v)

^{1/2}. The theorem then follows as c

^{2}= (v-w).(v-w) = v.v - 2 v.w + w.w = a

^{2}+ b

^{2}. This theorem is so important that it can be considered the ``fundamental theorem of geometry". Pythagoras teaching: From "The story of the greatest nations", (1910) From Archiv The Pythagoreans: Fyodor Bronnikov, (1827-1902)