## 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 X^{i}and Y^{i}is defined byBe 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 ?