·~ ...,... .chapter II 1 60 MULTILINEAR ALGEBRA Here a:"~ E v,' a:J E VJ for j =F i, and X EF. Thus, a function rjl: v 1 X ••• X v n --+ v is a multilinear mapping if for all i e { 1, ... , n} and for all vectors a: 1 e V 1> ... , a:1_ 1 eV1_h a:1+ 1eV1+ 1 , ••. , a:.ev•• we have rjl(a: 1 , ... ,a:1_ 1, ··, a 1+ 1, ... , a:.)e HomJ>{V" V). Before proceeding further, let us give a few examples of multilinear maps. Example 1. .2: Ifn o:= 1, then a function cf>: V 1 ..... Vis a multilinear mapping if and only if r/1 is a linear transformation. Thus, linear tra~sformations are just special cases of multilinear maps. 0 Example 1.3: If n = 2, then a multilinear map r/1: V 1 x V 2 ..... V is what we called .Multilinear Algebra a bilinear map in Chapter I. For a concrete example, we have w: V x v• ..... F given by cv(a:, T) = T(a:) (equation 6.6 of Chapter 1). 0 '\1 Example 1.4: The determinant, det{A), of an n x n matrix A can be thought of .l as a multilinear mapping cf>: F .. x · · · x P--+ F in the following way: If 1. MULTILINEAR MAPS AND TENSOR PRODUCTS a: (a , ... , a .) e F" for i 1, ... , o, then set rjl(a: , ... , a:.) det(a J)· The fact "'I 1 = 11 1 = 1 = 1 that rJ> is multilinear is an easy computation, which we leave as an exercise at the In Chapter I, we dealt mainly with functions of one variable. between vector end of this section. 0 · spaces. Those functions were linear in that variable and were called linear transformations. In this chapter, we examine functions of several variables Example 1.5: Suppose A is an algebra over F with multiplication denoted by between vector spaces. If s.uch a function is linear in each of its variables, then a:{J for a., {J eA. Let n ;;;?:: 2. We can then define a fun<'tion J.l: Ao --+A by the function is called a multilinear mapping. Along with any theory of multilinear J.t(a , ... , « ) = a: a: • .. a:n. Clearly J.l is a multilinear mapping. 0 ,. 1 0 1 2 maps comes a sequence of universal mapping problems whose solutions are the fundamental ideas in multilinear algebra. In this and the next few sections, we If rJ>: V 1 x .. · x Vo--+ Vis a ·multilinear map and Tis a linear transformation shall give a .:areful explanation of the pri.ncipal constructions of the subject from v tow, then clearly, TrJ>: v1 X •.• X vn--+ w is again a multilinear map. matter. Applications of the ideas discussed here will abound throughout the rest We can use this idea along with Example 1.5 above to give a few familiar of the book. examples from analysis. , ,j Let us first give a careful definition of a multilinear mapping. As usual, F will . denote an arbitrary field. Suppose V 1 , ... , V n and V are vector spaces over F. Example 1.6: Let I be an open interval in R. Set C"'(I) = n:-.1 ~x(I). Thus, X • • • X n X '· • X Let rJ>: V 1 V --+ V be a function from the finite product V 1 V 0 to C"'(I) consists of those f e C(I) such that f is infinitely differentiable on I. V. We had seen in Section 4 of Chapter I tltat a typical vector in V1 x · · · x V, i:s Clearly, C"'(l) is an algebra over IR when we define vector addition ann-tuple (a:1 , .•• , a:J with a:1 e V1• Thus, we can think of 4> as a function of then [(f + gXx) = f(x) + g(x)], scalar multiplication [(yf)(x) = yf(x)], and algebra variable vectors a: 1 , ••• , «a. multiplication [(fgXx) = f(x)g(x)] in the usual ways. Let D: C"'(I)-+ C"'(I) be the function that sends a given fe C.,(I) to its derivative f'. Thus, D(O =f. Clearly, Definition 1.1: A function rJ>: V 1 x · · · x Vn -> V is called a multilinear mapping De Homa(C"'(I), C""(I)). if for each • = 1, ... , n, we have Let n eN. Define a map rJ>: {C""(I)}"-+ C"'(I) by r/>(f1 , .. , f.) = D(f1 • .. fnl· Our comments immediately proceeding this example imply that 4> is a (a) r/>(a:1, ... , «t + ex;, ... , a:.) = r/>(a:l, .. · • «~o · .. , a:J + rJ>(a:1, · .. , a:;, ... , a:J, multilinear mapping. 0 and Example 1.7: Let [a, b] be a closed interval in Rand consider C([a, b]). Clearly, (b) r/>(a: 1, ... , xa1, ... , a:J = xr/>(«1, ... , «~o···· a:.). C([a, b]) is an R-algebra under the same pointwise operations given in Example 59 1.6. We can define a multilinear, real valued function 1/J: C([a, b])"--+ IR by I/J(f1, · .. ,f.)= J~fl '"fn. 0 MV~IIUNtAH MAI':i A NO TENSOR PRODUCTS til ~ Let us denote the collection of multilinear mappings from V 1 x · · · x V. to v by Mul~l X ... X v •• V). If = v1 X ... X v.,. then clearly, z MULTILINEAR ALGEBRA 2 62 MulJ{V 1 x · · · x V•• V) is a subset of the vector space V . In particUlar, if f, ge Mul~ x · · · x V.., V) and y e F, then xf yg is a vectorin A simple 1 x. + VZ. Question 1.9 is called the universal mapping problem for multilinear computation shows that xf + yg is in fact a multilinear mapping. This proves the mappings on v1 X ... X vn. In terms of commutative diagrams, the universal first assertion in the following theorem: problem can be stated as follows: Can we construct a multilinear map tf>: V 1 X · • · X Vn .... V with the property that for any multilinear map Theorem 1.8: Let V 1 , •••• V. anci V be vector spaces over F. Set Z = 1/1: V1 x ··· x v.--+ W there exists a unique TeHom,(V, W) such that the V1 x ··· x v •. Then following diagram is commutative: (a) x · · · x V ,., is a subspace of Mul~ 1 V) VZ. 1.10: (b) If n ;?; 2, {MuiJ<{V 1 X • • • X Va• V)} n {Hom~ 1 X • • • X V"' V)} = (0). ' lb Proof We need prove only (b). Suppose if,: V 1 x · · · x v ...... V is a multi­ I linear mapping that is also a linear transformation. Fix i = 1, .. : , n. v,x\/v Since ,P is multilinear, we have tP{a 1 , ••• , a.1 + a.1, ••• , a.) = tP{a 1 , ••• , a.J + tP{alt ... , a.). Since t/> is linear, we have t/>(a1 , . .. , a 1 + a.., ... , a.) = tf>(a 1 , • • • , a,. ... , a.J + tP(O, ... , a.,. ... , 0). Comparing the two results w gives tP{a1,. •• , a.)= t/>(0 , ... , a1, ••• , 0). Again since tP is linear, we have , ••• , , ••• , , ... , _ , tf>(a1 aJ=Ej. 1 ,P(O, ... , a1 Cl). Thus, ,P(a 1 a1 1 0, a 1u, Notice that a solution to 1.9 consists of a vector space V and a multilinear map • o • 1 = 0, a.J ,P: V x · · · x V. --+ V. The pair (V, ,P) must satisfy the following property: IfW is I Now n ;?; 2, the a are arbitrary, and so is the index i. Therefore, for any 1 1 any vector space over F and 1/1: V 1 x · · · x V.--+ W is·any multilinear mapping, (a , ... , a.)eV x ··· x v., we have ,P(a , ... , a.)= t/>(0, a , .•• , a.) I 1 1 1 2 then there must exist a unique, linear transformation T: V -+ W such that 1.10 +t/>(a1, 0, ... , 0) = 0 + 0 = 0. 0 commutes. I Before constructing a pair (V, ,P) satisfying the properties in 1.9, let us make Theorem .1.8(b) says that in general (i.e., when n;?; 2) a nonzero, multi­ the observation that any such pair is essentially unique up to isomorphism To linear mapping ,P: V1 x ... x v.-+ V is not a linear transformation from be more precise, we have the following lemma: V1 x ... x v. to V, and vice versa. We must always be careful not to confuse these tWO concepts When dealing with functions from Vl X ... X V tO V. 0 Lemma 1.11: Suppose (V, t/>) and (V', ,P') are two solution.q <o 1.9. Then there We have seen in Examples 1.6 and 1.7 that one method for constructing exist two isomorphisms T 1 E HomF(V, V') and T 2 e HomJ<{V', V) such that multilinear mappings on V 1 x · · · x V. is to choose a fixed multilinear map from V 1 x · · · x ' • to V and then compose it with various linear trans­ (a) T T = 1.-, T T = lv, and formations from V to other vector spaces. A natural question arises here. Can we 1 1 2 1 {b) the following diagram is commutative: construct {with possibly a judicious choice of V) all possible multilinear maps on Vi x · · · x V, by this method? This question has an affirmative answer, which 1.12: leads us to the construction of the tensor product of V 1 , ... , V". Before proceeding further, let us give a precise statement of the problem we wish to consider. • v v,x~ / , . • , 1.9: Let V 1 V. be vector spaces over F. Is there is a vector space V {over F) and a multilinear mapping ,P: V 1 x · · · x V" ...