Algebra multilinear pdf

Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis.

The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms.

One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. Section 7 begins with the tensor product of three and then n vector spaces, carefully considering the universal mapping property and the question of associativity.

The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. This is an associative algebra with a universal mapping property relative to any linear mapping of E into an associative algebra A with identity: the linear map extends to an algebra homomorphism of T E into A carrying 1 into 1.

Sections 8—9 define the symmetric and exterior algebras of a vector space E. The symmetric algebra S E is a quotient of T E with the following universal mapping property: any linear mapping of E into a commutative associative algebra A with identity extends to an algebra homomorphism of S E into A carrying 1 into 1.

The symmetric algebra is commutative. Similarly the exterior algebra? E extends to an algebra homomorphism of? E into A carrying 1 into 1.

The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras.

Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions. Unable to display preview. Download preview PDF. If there is a survey it only takes 5 minutes, try any survey which works for you. Multilinear Algebra. File Name: Multilinear Algebra. Download Now! Register a free Trial Account.

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