Looking back to (36), we have for the dual matrix (Ā[function]*A) (A^{-1}[function]A) = A^{-1}[function]*[function]A = Det^{1/2} function. A^{-1}A = Det^{1/2}[function] from which it is to be seen that the dual matrix [function]* behaves exactly like the primary matrix [function], and is therefore a space time vector of the II kind; [function]* is therefore known as the dual space-time vector of [function] with components ([function]_{1 4}, [function]_{2 4}, [function]_{3 4},), ([function]_{2 3̄}, [function]_{3 1}, [function]_{1 2}).
6.* If w and s are two space-time rectors of the 1st kind then by w [=s] (as well as by s [=w]) will be understood the combination (43) w_{1} s_{1} + w_{2} s_{2} + w_{3} s_{3} + w_{4} s_{4}.
In case of a Lorentz transformation A, since (wA) (Ā[=s]) = w s, this expression is invariant.—If w [=s] = 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
| w_{1} w_{2} w_{3} w_{4} |
| s_{1} s_{2} s_{3} s_{4} |
Then it follows immediately that the system of six magnitudes (44) w_{2} s_{3} - w_{3} s_{2}, w_{3} s_{1} - w_{1} s_{3}, w_{1} s_{2} - w_{2} s_{1}, w_{1} s_{4} - w_{4} s_{1}, w_{2} s_{4} - w_{4} s_{2}, w_{3} s_{4} - w_{4} s_{3},
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^{1/2}[w, s] = 0. The dual vector of [w, s] shall be written as [w, s].*
If [=w] is a space-time vector of the 1st kind, [function] of the second kind, w [function] signifies a 1 × 4 series matrix. In case of a Lorentz-transformation A, w is changed into w´ = wA, [function] into [function]´ = A^{-1} [function] A,—therefore w´ [function]´ becomes = (wA A^{-1} [function] A) = w [function] A i.e. w [function] is transformed as a space-time vector of
