tensor_products math 113

Math 113: Tensor Products 1. Fundamental properties This past week, you proved some rst properties of the tensor product V Wof a pair of vector spaces VandW. This week, I want to rehash some fundamental properties of the tensor product, that you you are welcome to take as a working denition from here forwards. LetVandWbe two nite-dimensional vector spaces over F. The tensor product V W is a vector space over F, equipped with a map :VW!V W: We often refer to the vector (v;w) inV Was the vector v w. The vector space V W and the map satisfy the following fundamental properties: 1. The map isbilinear . (Why should it be bilinear? Recall that ordinary multiplication of real numbers ( x;y)7!x
yis a bilinear map, so this is one generalization of multiplying). We sometimes call vector multiplication . Call any vector in the image of (any element inV Wof the form v w) apure tensor . By bilinearity of , we see that (av+bv0) w=(av+bv0;w) =a(v;w) +b(v0;w) =av w+bv0 w; and similarly v (aw+bw0) =av w+bv w0: 2. If ( v1;:::;vn) is any basis of Vand (w1;:::;wm) is any basis of W, then the collection fvi wj=(vi;wj);1in;1jmg is always a basis of V W. Thus, (1) dim( V W) = dimV
dimW: 3. Not every element in V Wis necessarily of the form v w. One way to see this is to work in terms of a basis: note that given any pair ( v;w), we can write them in terms of a basis as v=a1v1+


+anvnandw=b1w1+


+bmwm, for some scalars a1;:::;an, b1;:::;bm. This implies that (2) (v;w) =v w=nX i=1mX j=1ai
bjvi wj: 1 But a general element in V Wis of the form nX i=1nX j=1cijvi wj; wherecijis a collection of mnscalars. Such a collection cannot always be written as ai
bjfor a collection of m+nscalarsa1;:::;an;b1;:::;bm. This is equivalent to the fact that not every polynomial in two variables x,ycan be written as the product (a0+a1x+


+anxn)
(b0+b1y+


+bnym). For example, xy+1 cannot be written this way. We conclude that while every element in V Wis a sum of elements of the form v w, not every element is necessarily of this form (unless the dimension of either Vor Wis 0 or 1). 4. On the past homework, you proved, or attempted to prove, the following proposition: Proposition 1.1.IfF:VW!Xis any bilinear map, then Fcan be written uniquely as a composition VW!V WF!X; whereis the multiplication map, and Fis a linear map. Let's rst note we already know what F:V Whas to be on pure tensors: F(v w) =F((v;w)) =F(v;w) Since a general vector in V Wis a sum of some pure tensors, and we want Fto be linear, it is determined by its value on pure tensors (extending to all of V Wby linearity). The proposition guarantees that the result Fis indeed a linear map. Corollary 1.1.LetL(VW;X )denote the space of bilinear maps from VWinto X. Then, there is an isomorphism L(VW;X )=L(V W;X ) where the right hand side denotes the space of linear maps out of the tensor product. The above proposition has important ramications, some of which you explored on HW 7. Crucially for us, it will give us a way of constructing linear maps out of the tensor product. Constructing linear maps T:V W!X: The above discussion gives us two ways of constructing linear maps out of the tensor product of vector spaces: Just dene the values of the map on a basis fvi wjg, and extend by linearity; or Dene the Ton pure tensors, of the form v w, and extend by linearity to all of V W. By part 4 above, as long as the map ^T:VW!X (v;w)7!T(v w) is bilinear, you are guaranteed that Tis linear! . 2 2. First examples, and applications Given the intimate relationship of tensor products to bilinear maps (property 4), it should not be surprising that tensor products arise whenever there are bilinear maps lurking around. In this section, we'll see a few examples of where tensor products might begin to arise, and examples of tensor products. (1)Rm Rn=Rmn. It might be fairer to say that Rm Rnis isomorphic to the space Mat( m;n;R) ofmnreal matrices, which also has dimension mn. The correspondence is straightforward to describe: Rm Rnhas a basis coming from the pairs of standard basis vectors: fei ej;1im;1jn; and Mat(m;n;R) has the standard basis eij;1im;1jn whereeijis the matrix that has a 1 in component ( i;j) and 0 elsewhere. One choice of isomorphism is the identication ei ej7!eij: (2) Let F[x]ndenote the vector space of polynomials in xwithFcoecients, and degree n. LetF[y]mdenote the vector space of polynomials in ywithFcoecients and degreem. And let F[x;y]n;mdenote the vector space of polynomials in two variablesx;ywithFcoecients, such that the degree of xand the degree of yin any term arenandmrespectively. Then, there is an isomorphism F[x]n F[y]m!F[x;y]n;m given by identifying the basis element xi yj of the left hand side with the basis element xiyj on the right hand side. (3)Inner products . An inner product structure on a real vector space Vis a bilinear map F=h
;
i:VV!R satisfying some additional conditions (positivity, deniniteness, symmetry). By 4 in the previous section, an inner product structure induces a linear map F:V V!R; i.e., an element F2(V V): 3 Positivity, denitiness, and symmetry then translate into properties of F. This is occasionally a useful perspective. (4)Relation to spaces of linear maps . Given that V Whas the same dimension as L(V;W ), one might expect some sort of relationship. There is not a canonical one, but there is a canonical relationship between V WandL(V;W )! (Vhas the same dimension as V, as we know, so these two spaces have the same dimension too). On homework this week, you will construct a canonical linear map V W!L (V;W ) (To do this: recall the advice given in the previous section about constructing linear maps out of the tensor product). This map is an isomorphism whenever VandW are nite dimensional. We'll use this correspondence on Homework 8 to give a basis-free denition of thetrace of a linear map. 4