7. 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 :

 

(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 ?