Abstract
Readers likely recall the following elementary geometric statement: if the sides of an angle α in a Euclidean plane are orthogonal to the sides of an angle β, then α=β or α+β=180°; in other words, |cos α|=|cos β|. A multidimensional generalization of this theorem relates to an interaction of skew symmetry and orthogonality features, i.e., properties of determinants and other multilinear functions, and Euclidean spaces; we presume readers are familiar with these features within the scope of a common university course, and on this basis we develop the tools necessary to prove this generalization. These tools have multiple applications. For examples, they allow us to extend a definition of the angle between straight lines or between hyperplanes to k-dimensional planes of ℝn for every 0<k<n and to extend a definition of the vector product of two vectors in ℝ3−to any number of vectors and multivectors in ℝn. Readers may learn more about the by consulting the guide to the literature provided in this chapter.
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Notes
- 1.
Readers experienced in abstract algebra know that exterior products and other multilinearity-related techniques are developed in algebra for free modules over associative–commutative unitary rings. However, here we do not need such a level of generality.
- 2.
Recall that the dual space L * for a vector space L is a vector space of all linear functionals in L.
- 3.
Supplied with determinate topological and manifold structures, which we will not consider.
- 4.
Readers familiar with projective geometry know that P(∧kℝn) is referred to as a projectivization of the vector space ∧k ℝn.
- 5.
A linear map \( L\mathop{\to}\limits^D{L^{*}} \) defines the bilinear forms \( {{\varPhi^{\prime}}_D}\left( {x,y} \right):=Dx(y) \), \( {{\varPhi^{\prime\prime}}_D}\left( {x,y} \right):=Dy(x) \). Readers experienced in abstract algebra know that the correspondences \( D\mapsto {{\varPhi^{\prime}}_D} \), \( \varPhi \mapsto {{D^{\prime}}_{\varPhi }} \) and \( D\mapsto {{\varPhi^{\prime\prime}}_D} \), \( \varPhi \mapsto {{D^{\prime\prime}}_{\varPhi }} \) define two pairs of functorial isomorphisms Hom (L,L *) ⇄ L * ⊗ L * between vector spaces of linear operators L → L * and bilinear forms in L.
- 6.
Readers preferring more developed terminology may say that orienting ℝn is choosing one of two connected components of the linear group GL(n,ℝ) (see section P6.8*** in the “Polar and Singular Value Decomposition Theorems” problem group above).
- 7.
The closure, with respect to multiplications by rational scalars, follows from the closure with respect to summations (why?), but for real scalars it does not. (Provide examples.)
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Roytvarf, A.A. (2013). A Property of Orthogonal Matrices. In: Thinking in Problems. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8406-8_8
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DOI: https://doi.org/10.1007/978-0-8176-8406-8_8
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