(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
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
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