ABSTRACT
This last chapter is devoted to tensor products of vector spaces and related topics such as the symmetric and exterior algebras. The name, tensor product, arises from its applications in differential geometry where it may be applied to the tangent or cotangent space of a manifold, but its utility is broadly felt throughout mathematics. For example, in group theory, tensor products are used to construct group representations and, in many contexts, the tensor product is used to extend the base field of a vector space, for example, from the real field to the field of complex numbers. In the first section, we define the tensor product of vector spaces as the solution to a certain universal mapping problem and prove that it exists. In the second section, we make use of the definition of the tensor product to prove some “functorial” properties, such as how the tensor product behaves with respect to direct sums. We show how a tensor product of linear transformations can be defined to obtain a transformation from one tensor product to another. Finally, we investigate how to compute the matrix of a tensor product of transformation from the matrices of the transformations. In section three, we use the tensor product to construct a universal algebra for a given vector space V. In the final section, we introduce the notion of a Z-graded algebra and related concepts such as a homogeneous ideal. We apply these ideas to the tensor algebra and construct the symmetric algebra and the exterior algebra of a vector space as quotient spaces of the tensor algebra by particular homogeneous ideals. We show that the symmetric algebra of a vector space of dimension n over a field F is isomorphic to the algebra of polynomials in n-variables. We show that the symmetric algebra and the exterior algebra are both solutions to universal mapping problems. Finally, we show how a linear transformation from a vector space V to a vector space W induces a linear transformation on the exterior algebra and its homogeneous pieces.
