7. Two tiny Proofs using Matrix Algebra
7.1 Proof of the Transformation Rule for the Dual Forms of Four-Vectors:
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 ?