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 Det^{½} (Ā _f_ A) = Det A. Det^{½} _f_. Therefore Det^{½} _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 (Ā_f_*A) (A⁻¹_f_A) = A⁻¹_f_*_f_A = Det^{½} function. A⁻¹A = Det^{½}_f_ 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 (_w_A) (Ā_ṡ_) = _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_, _w₃_ _s₁_ - _w₁_ _s₃_, _w₁_ _s₂_ - _w₂_ _s₁_, _w₁_ _s₄_ - _w₄_ _s₁_, _w₂_ _s₄_ - _w₄_ _s₂_, _w₃_ _s₄_ - _w₄_ _s₃_, 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 Det^{½} [_w_, _s_] = 0. The dual vector of [_w_, _s_] shall be written as [_w_, _s_]. If _ẇ_ is a space-time vector of the 1st kind, _f_ of the second kind, _w_ _f_ signifies a 1 × 4 series matrix. In case of a Lorentz-transformation A, _w_ is changed into _w′_ = _w_A, _f_ into _f′_ = A⁻¹ _f_ A,—therefore _w′_ _f′_ becomes = (_w_A A⁻¹ _f_ A) = _w_ _f_ A _i.e._ _w_ _f_ is transformed as a space-time vector of the 1st kind.[23] We can verify, when _w_ is a space-time vector of the 1st kind, _f_ of the 2nd kind, the important identity (45) [_w_, _w__f_] + [_w_, _w__f_*]* = (_w_] _ẇ_)_f_. The sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices. For example, for ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_, ω_f_ = | _i__f_₄₁, _i__f_₄₂, _i__f_₄₃, 0 |; ω_f_* = | _i__f_₃₂, _i__f_₁₃, _i__f_₂₁, 0 | [ω · ω_f_] = 0, 0, 0, _f_₄₁, _f_₄₂, _f_₄₃; [ω · ω_f_*]* = 0, 0, 0, _f_₃₂, _f_₁₃, _f_₂₁.