M7   Two tiny Proofs using Matrix Algebra

7.1 Proof of the Transformation Rule for the Dual Forms of Four-Vectors:

  We have

 7.2 Proof of the Lorentz-Invariance of the STR Scalar Product:

The scalar product of two four-vectors Xⁱ and Yⁱ is defined by

Be aware of the notation without a dot between the four-vectors !

This is easily expressed by a matrix multiplication of a four-form with a four-vector :

mmmmmmm
mmmmmmm

(No dot on the left side, a dot designating the matrix multiplication on the right side)

The definition of four-vectors and 7.1  yield the general Lorentz-invariance of this product: 

  

  

 

Easy, isn't it ?

M and F are more than nice Further Improvements