Note that there are two pieces of data in a tensor product. While we do not discuss general differentiable manifolds, we do include a chapter on vector and tensor fields defined on hypersurfaces in a euclidean manifold. Positions are not absolute they must be measured relative to a reference point. The tensor product of two modules a and b over a commutative ring r is defined in exactly the same way as the tensor product of vector spaces over a field. A tensor product of xand yis a vector space zover k, together with a bilinear map. Let v and w be vector spaces over a eld k, and choose bases fe igfor v and ff jgfor w. In preparing this two volume work our intention is to present to engineering and science students a modern introduction to vectors and tensors. The state of such a system is a unit vector in a kdimensional.
Tensor products of vector spaces we can use the same process to define the tensor product of any two vector spaces. C,\otimes with a right and left action, respectively, of some monoid a a, their tensor product over a a is the quotient of their tensor product in c c by this action. On tensor products, vector spaces, and kronecker products. Chapter 22 tensor algebras, symmetric algebras and exterior. Here are the main results about tensor products summarized in one theorem.
This chapter describes the several theorems and propositions related to the tensor product of vector spaces. This volume contains frequent references to volume 1. Tensor algebras, symmetric algebras and exterior algebras 22. Particular tensor products are those involving only copies of a given vector space v and its dual v these give all the tensors associated to the. The aim of my master project is to study several tensor products in riesz space theory and in particular to give new constructions of tensor products of integrally closed directed partially ordered vector spaces, also known as integrally closed preriesz spaces, and of banach lattices without making use of the constructions by d. Tensor products first arose for vector spaces, and this is the only setting where they occur in physics and engineering, so well describe tensor products of vector. Nov 18, 2018 so a tensor product is like a grownup version of multiplication. Most of the time, the vectors will be shown in the ket notation, like j i. Instead, the construction below of the tensor product v rc complexi cation of v of a real vectorspace v with c over r is exactly right, as will be discussed later.
Overview 5 overview and summary this monograph is meant as a user guide for both tensor products and wedge products. Elementary tensor products a tensor product of two vectors is an outer product that. Suppose that for every bilinear map f defined on uxv there is a. The tensor product of two vector spaces is a new vector space with the property that bilinear maps out of the cartesian product of the two spaces are equivalently linear maps out of the tensor product the tensor product of vector spaces is just the special case of the tensor product of modules over some ring r r for the case that this ring happens to be a field. The tensor product of vector spaces makes the category vect of all vector spaces into a monoidal category, in fact a distributive monoidal category. If v1 and v2 are any two vector spaces over a field f, the tensor product is a bilinear map. Tensor products of vector spaces are to cartesian products of sets as direct sums of vectors spaces are to disjoint unions of sets. W is the complex vector space of states of the twoparticle system. Introductiontovectorspaces,vector algebras,andvectorgeometries richard a. The eld f is a vector space over itself, with its usual operations. The term tensor product has many different but closely related meanings in its original sense a tensor product is a representing object for a suitable sort of bilinear map and multilinear map.
This is a beginners question on what exactly is a tensor product, in laymens term, for a beginner who has just learned basic group theory and basic ring theory. To volume 2 this is the second volume of a twovolume work on vectors and tensors. A vector in the primary space can be converted to a vector in the conjugate dual space and vice versa by the operation of transposition. Smith october 14, 2011 abstract an introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. A basis for the tensor product is all products of basis elements in one space. If a a is commutative, then this is a special case of the tensor product in a multicategory. The most classical versions are for vector spaces modules over a field, more generally modules over a ring, and even more generally algebras over a commutative monad. The concept of tensor product will be discussed using the examples of the tensors s, t and u of order 2, 3, and 2, respectively, knowing that the argument can be generalized to any ntuple tensors of any order.
Its what happens when you systematically multiply a bunch of numbers together, then organize the results into a list. The free vector space over f generated by ais the vector space freea consisting of all formal nite linear combinations of elements of a. A gentle introduction to tensors washington university in. Even in fairly concrete linear algebra, the question of extension of scalars to convert a real vector space to a complex vector space is possibly mysterious. Vector addition a quantity related to the displacement vector is the position vector for a point. These objects are sometimes glossed over in literature that makes heavy use of them, the assumption being that everything. The tensor product of vector spaces is just the special case of the tensor product of modules over some ring r r for the case that this ring happens to be a field. You can see that the spirit of the word tensor is there. The foundation of tensor products lies in a natural extension of the notion of linear map. In propositions 1,2,3, we assume given a tensor product h k. Chapter 4 has four parts involving products of two vectors and their vector spaces.
There is a natural isomorphism of vector spaces mapping to consequently, for any and the element can be regarded as a tensor of type and is called the tensor product of and. Tensor products of archimedean partially ordered vector spaces. Note that the three vector spaces involved arent necessarily the same. What these examples have in common is that in each case, the product is a bilinear map. Introduction to vectors and tensors vector and tensor analysis volume 2 ray m. This is a common occurrence when studying the injective and projective tensor products of functionsequence spaces and tvss.
The chapter mentions only vector spaces over the field c of complex numbers and assumes e and f as two vector spaces. The chapter also proves the existence of a tensor product of any two vector. Chapter 22 tensor algebras, symmetric algebras and. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.
Dimensionality of a vector space and linear independence. The tensor product let us start by constructing the tensor product v w. If v 1 and v 2 are any two vector spaces over a eld f, the tensor product is a bilinear map. We now show two nonzero elementary tensors in vector spaces are equal only under the \obvious circumstances. On tensor products, vector spaces, and kronecker products we begin with the denition of the tensor product. Of course we may be working in a vacuumit is conceivable that the axioms for a tensor product are inconsistent or only sometimes consistent. The tensor product of two rmodules is built out of the examples given above. Because it is often denoted without a symbol between the two vectors, it is also referred to as the open product. The graphics look ratty in windows adobe pdf viewers when not scaled up, but look just fine in this. B is the free rmodule generated by the cartesian product and g is the rmodule generated by the same relations as above. A short introduction to tensor products of vector spaces. As a result, we need learn something about the hilbert spaces for the purpose of learning quantum mechanics. A vector space v is a collection of objects with a vector.
Given two vector spaces, there is a unique algebraic notion of a tensor product, which is. After completing this task, we will brie y translate our result to the language of category theory. Thus a0 a iei is formed via the conversion e i ei whereas b b je j is formed via the conversion ej e j. Tensor product of vector spaces c if m1,1 and m2,2 are two tensor products of e and f, then there is a bijective linear map u such that the following diagram is commutative. This is a way of getting a new big vector space from two or more smaller vector spaces in the simplest way one can imagine.
The components of the product are computed according to the formula let, and let the numbers and be fixed with and. And then show that for finitedimensional spaces, you only need sums of boundedly many terms. The notion of extension of scalars has important senses in situations which are qualitatively di erent than complexi cation of real vector spaces. The purpose of x2 is to show that a tensor product always does exist. Pairings of tensor products we begin with the case of tensor products, as all others will be easily deduced from it after we have done the hard work in this case. Volume 1 is concerned with the algebra of vectors and tensors, while this volume is concerned with the geometrical. It is this chapter that provides the foundations for tensor applications in physics.
Uxvx be a bilinear map from uxv to a vector space x. It is also called kronecker product or direct product. In fact, rs is the smallest submodule that contains s. Hilbert spaces, tensor products this lecture will formalize many of the notions introduced informally in the second lecture. To give a construction of the tensor product, we need the notion of a free vector space. The tensor product is just another example of a product like this. The tensor product is a space which allows us to replace some bilinear more generally multilinear maps by linear maps. For instance, the components of a vector and of a covector coincide with their usual coordinates with respect to the bases and. Notes on locally convex topological vector spaces 5 ordered family of.
Tensor products rst arose for vector spaces, and this is the only setting where they occur in physics and engineering, so well describe tensor products of vector spaces rst. The multiplication is given by concatenating such symbols, imposing the distributive law under addition, and requiring that scalar multiplication commute with the tensor product. Tensorproduct spaces the most general form of an operator in h 12 is. The chapter also proves the existence of a tensor product of any. Tensor on a vector space encyclopedia of mathematics. This talk discusses a peculiar issue arising in functional analysis. Just to make the exposition clean, we will assume that v and w are. A dyad is a special tensor to be discussed later, which explains the name of this product. A vector space also called a linear space is a collection of objects called vectors, which may be added together and multiplied scaled by numbers, called scalars. The third chapter extends tensor theory to spaces other than vector spaces, namely manifolds.
The important thing is that it takes two quantum numbers to specify a basis state in h 12 a basis that is not formed from tensorproduct states is an entangledstate basis in the beginning, you should. Then there is a unique vector space aover kand a bilinear map f. The linear structures are genuine vector spaces or vector spaces with additional binary operations. A useful lemma about the tensor product is that it is unique, in the following sense. Tensor products, wedge products and differential forms. We say that t satisfies the char acteristic property of the tensor product with respect to v and w if there is a bilinear.