Module VI·Article II·~1 min read
Exterior Algebra and Determinant
Tensor Algebra
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Exterior Product
Exterior algebra $\Lambda V$ is an algebra with anticommutative multiplication: $v \wedge w = -w \wedge v$, and in particular $v \wedge v = 0$.
$\Lambda^k(V)$ — the space of $k$-forms of dimension $C(n, k)$.
Basis of $\Lambda^2(\mathbb{R}^n)$: ${e_i \wedge e_j : i < j}$. Basis of $\Lambda^n(\mathbb{R}^n)$: one-dimensional.
Determinant via Exterior Algebra
The determinant is the unique (up to scalar multiplication) $n$-form on an $n$-dimensional space:
$ \det A = \frac{e_1 \wedge \ldots \wedge e_n(Ae_1, \ldots, Ae_n)}{(e_1 \wedge \ldots \wedge e_n)(e_1, \ldots, e_n)}. $
Antisymmetry in rows and multilinearity — the axioms of the determinant — are simply properties of the exterior product.
Orientation
Choosing the orientation of an $n$-dimensional space means choosing one of two classes of ordered bases (based on the sign of the transition matrix determinant).
Change of variables in an integral: $|\det J|$ is the “volume scaling,” while $\operatorname{sign} \det J$ is the preservation or reversal of orientation.
Application in Geometry
The $k$-dimensional volume of a parallelepiped spanned by $v_1,\ldots,v_k$ in $\mathbb{R}^n$: $\sqrt{\det(G^\mathrm{T}G)}$, where $G = [v_1|\ldots|v_k]$ (Gram matrix).
This is a generalization: for $k=1$ it is the norm of the vector, for $k=2$ — the area of a parallelogram, for $k=n$ — the absolute value of the determinant.
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