(37), we have the identity . Therefore becomes an invariant in the case of a Lorentz transformation [see eq. (26) Sec. § 5].

Looking back to (36), we have for the dual matrix

from which it is to be seen that the dual matrix behaves exactly like the primary matrix *f*, and is therefore a space time vector of the II kind; is therefore known as the *dual space-time vector* of *f* with components .

6°.If *w* and *s* are two space-time vectors of the 1st kind then by (as well as by ) will be understood the combination

(43) |

In case of a Lorentz transformation , since this expression is invariant. — If , then *w* and *s* are *perpendicular* to each other.

Two space-time rectors of the first kind *w, s* gives us a 2✕4 series matrix

Then it follows immediately that the system of six magnitudes

(44) |

behaves in case of a Lorentz-transformation as a space-time vector of the II. kind. The vector of the second kind with the components (44) are denoted by *[w,s]*. We see easily that . The dual vector of *[w,s]* shall be written as *[w,s]**.

If *w* is a space-time vector of the 1st kind, *f* of the second kind, *wf* signifies a 1✕4 series matrix. In case of a Lorentz-transformation , *w* is changed into , *f* into , therefore , *i.e.,* *wf* is transformed as a space-time vector of the 1st kind. We can verify, when *w* is a space-time vector of the 1st kind, *f* of the 2nd kind, the important identity

(45) | . |