|f₄₁ f₄₂ f₄₃ 0 |
and is to be replaced by A⁻¹ f A in case of a Lorentz transformation [see the rules in [§ 5] (23) (24)]. Therefore referring to the expression (37), we have the identity Det½ (Ā f A) = Det A. Det½ f. Therefore Det½ f becomes an invariant in the case of a Lorentz transformation [see eq. (26) See. [§ 5]].
Looking back to (36), we have for the dual matrix (Āf*A) (A⁻¹fA) = A⁻¹f*fA = Det½ function. A⁻¹A = Det½f from which it is to be seen that the dual matrix f* behaves exactly like the primary matrix f, and is therefore a space time vector of the II kind; f* is therefore known as the dual space-time vector of f with components (f₁₄, f₂₄, f₃₄,), (f₂₃}, f₃₁, f₁₂).
6. If w and s are two space-time rectors of the 1st kind then by w ṡ (as well as by s ẇ) will be understood the combination (43) w₁ s₁ + w₂ s₂ + w₃ s₃ + w₄ s₄.
In case of a Lorentz transformation A, since (wA) (Āṡ) = w s, this expression is invariant.—If w ṡ = 0, 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
| w₁ w₂ w₃ w₄ |
| s₁ s₂ s₃ s₄ |
Then it follows immediately that the system of six magnitudes (44)
w₂ s₃ - w₃ s₂,