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The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x. All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring .
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.
A Cayley graph of the symmetric group S 4 using the generators (red) a right circular shift of all four set elements, and (blue) a left circular shift of the first three set elements. Cayley table, with header omitted, of the symmetric group S 3. The elements are represented as matrices. To the left of the matrices, are their two-line form.
Formally, the symmetric algebra of a vector space V over a field F is the group algebra of the dual, Sym(V) := F[V ∗], and the Weyl algebra is the group algebra of the (dual) Heisenberg group W(V) = F[H(V ∗)]. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V).
A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any m × n {\displaystyle m\times n} matrix.
The tensor algebra of a vector space V. The homogeneous elements of degree n are the tensors of order n , T n V {\displaystyle T^{n}V} . The exterior algebra ⋀ ∙ V {\displaystyle \textstyle \bigwedge \nolimits ^{\bullet }V} and the symmetric algebra S ∙ V {\displaystyle S^{\bullet }V} are also graded algebras.