Retrieving "Determinant" from the archives

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1. Lattice Vector

   Linked via "determinant"

   $$V = |\mathbf{a}1 \cdot (\mathbf{a}2 \times \mathbf{a}_3)|$$
   The choice of basis vectors is non-unique. If $\{\mathbf{a}i\}$ is a valid basis, then $\{\mathbf{a}i'\}$ is also a valid basis if the transformation matrix $M$ relating them has a determinant of $\pm 1$.
   $$\mathbf{a}i' = \sumj M{ij} \mathbf{a}j \quad \text{where} \quad \det(M) = \pm 1$$

2. Lorentz Group

   Linked via "determinant"

   This choice of metric dictates the $(+,-,-,-)$ signature convention.
   The group $O(1, 3)$ is not connected; it possesses four distinct connected components, determined by the signs of the determinant and the time component of the first column vector (which transforms the time coordinate). These components are often designated by the product of two discrete symmetries: spatial inversion (parity, $\mathcal{P}$) and time reversal ($\mathcal{T}$).
   The connected component containing the […