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Mathematics-Online course: Linear Algebra - Basic Structures - Scalar Product and Norm

Pythagorean Theorem

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If $ u\perp v$, then we have

$\displaystyle \vert u+v\vert^2 = \vert u\vert^2 + \vert v\vert^2\, . $

(Authors: Burkhardt/Höllig/Hörner)
The following holds true:

$\displaystyle \vert u+v\vert^2=\langle u+v,u+v \rangle = \langle u,u\rangle + ... ...since } v \perp u} +\langle v,v\rangle = \vert u\vert^2+\vert v\vert^2\,. $

Today this theorem is known as the Pythagorean theorem, although the Babylonians already knew it some 1000 years earlier, but Pythagoras might have been the first to prove it.

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005