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Explain the graded associative algebra ∧V in the exterior algebra of a vector space V, and its corresponding exterior product.
Explain the graded associative algebra ∧V in the exterior algebra of a vector space V, and its corresponding exterior product.
The graded associative algebra ∧V in the exterior algebra of a vector space V is defined as $∧ V = k ⊕ V ⊕ ∧^2 V ⊕ ∧^3 V ⊕ ⋯$. The corresponding product ∧ on the exterior algebra ∧V is called the exterior product or wedge product.
Define a k-blade in the context of the exterior algebra.
Define a k-blade in the context of the exterior algebra.
A k-blade in the exterior algebra corresponds to the oriented parallelotope spanned by elements of the form v1∧...∧vk, where vi ∈ V.
What are k-multivectors in the exterior algebra, and how are they represented?
What are k-multivectors in the exterior algebra, and how are they represented?
In the exterior algebra, elements in ∧kV are called k-multivectors and are given by a sum of k-blades. They are an abstraction of oriented lengths, areas, volumes, and more generally oriented k-volumes for k ≥ 0.
State and explain the alternating property of the exterior product ∧ in the exterior algebra.
State and explain the alternating property of the exterior product ∧ in the exterior algebra.
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Why is the exterior algebra also called the Grassmann algebra, and who is it named after?
Why is the exterior algebra also called the Grassmann algebra, and who is it named after?
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