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Multilinear Algebra Appendix A Multilinear Algebra This chapter presents concepts from multilinear algebra based on the basic properties of finite dimensional vector spaces and linear maps. The primary aim of the chapter is to give a concise introduction to alternating tensors which are necessary to define differential forms on manifolds. Many of the stated definitions and propositions can be found in Lee [1], Chaps. 11, 12 and 14. Some definitions and propositions are complemented by short and simple examples. First, in Sect. A.1 dual and bidual vector spaces are discussed. Subsequently, in Sects. A.2–A.4, tensors and alternating tensors together with operations such as the tensor and wedge product are introduced. Lastly, in Sect. A.5, the concepts which are necessary to introduce the wedge product are summarized in eight steps. A.1 The Dual Space Let V be a real vector space of finite dimension dim V = n.Let(e1,...,en) be a basis of V . Then every v ∈ V can be uniquely represented as a linear combination i v = v ei , (A.1) where summation convention over repeated indices is applied. The coefficients vi ∈ R arereferredtoascomponents of the vector v. Throughout the whole chapter, only finite dimensional real vector spaces, typically denoted by V , are treated. When not stated differently, summation convention is applied. Definition A.1 (Dual Space)Thedual space of V is the set of real-valued linear functionals ∗ V := {ω : V → R : ω linear} . (A.2) The elements of the dual space V ∗ are called linear forms on V . © Springer International Publishing Switzerland 2015 123 S.R. Eugster, Geometric Continuum Mechanics and Induced Beam Theories, Lecture Notes in Applied and Computational Mechanics 75, DOI 10.1007/978-3-319-16495-3 124 Appendix A: Multilinear Algebra The dual space, equipped with pointwise addition and scalar multiplication is again a real vector space. 1 n ∗ Proposition A.2 Given any basis (e1,...,en) for V , let ε ,...,ε ∈ V be the linear forms defined by εi ( ) := δi , e j j (A.3) δi where j is the Kronecker delta symbol defined by 1ifi = j δi := (A.4) j 0ifi = j . 1 n ∗ Then (ε ,...,ε ) is a basis for V , called the dual basis to (e1,...,en) and conse- quently dim V ∗ = dim V . Any linear form ω can be uniquely represented as a linear combination i ω = ωi ε , (A.5) with components ωi = ω(ei ). Proof We need to show that (ε1,...,εn)(i) spans the dual space V ∗ and (ii) is linearly independent. Let v ∈ V and ω ∈ V ∗. (i) Due to the linearity of ω we have (A.1) i (A.2) i ω(v) = ω(v ei ) = v ω(ei ). (A.6) Writing the linear form as a linear combination (A.5) and applying it to v, it follows by linearity of the linear form and the definition of the dual basis (A.3) that ω( ) = ω εi (v j ) (A=.2) ω v j εi ( ) (A=.3) ω v j δi (A=.4) ω vi . v i e j i e j i j i (A.7) 1 n Let ωi = ω(ei ). Then comparison of (A.6) and (A.7) proves that (ε ,...,ε ) spans V ∗. (ii) To show that (ε1,...,εn) is linearly independent, suppose some linear com- i bination equals zero, i.e. ω = ωi ε = 0. Applying to both sides an arbitrary vector v ∈ V , it follows by the computation of (A.7) that i i ∀v ∈ V : ω(v) = 0 ⇒∀v ∈ R: v ωi = 0 ⇒ ωi = 0 . Thus, the only linear combination of elements of (ε1,...,εn) that sums to zero is the trivial one. This proves the linear independency of (ε1,...,εn). The application of a linear form ω ∈ V ∗ on a vector v ∈ V is called the duality pairing and is expressed in components as in (A.7). Appendix A: Multilinear Algebra 125 Since the dual space V ∗ is also a vector space, we may consider the dual space of V ∗, called the bidual space V ∗∗ := (V ∗)∗. For each vector space V , there exists a linear isomorphism ∗∗ ∗ φ : V → V , v → φv :∀ω ∈ V ,φv(ω) = ω(v). Hence, we identify the vectors of the bidual space V ∗∗ naturally with the vectors of the vector space V . For convenience, we suppress the function φ in our notation and write φv(ω) =: v(ω). ∗∗ The duality pairing between the identified bidual basis ei ∈ V and the dual basis ε j ∈ V ∗ is evaluated as (ε j ) = δ j . ei i A.2 Multilinear Forms and Tensors Definition A.3 (Multilinear Form, Tensor) Suppose V1,...,Vk are vector spaces. AmapF: V1 ×···×Vk → R is said to be multilinear, if it is linear in each argument, i.e. for any i ∈ {1,...,k}, v j ∈ Vj , wi ∈ Vi , a, b ∈ R F(v1,...,avi +bwi ,...,vk) = aF(v1,...,vi ,...,vk)+bF(v1,...,wi ,...,vk). We refer to multilinear F as multilinear form or tensor of rank k.1 The set of such multilinear forms is denoted by L(V1,...,Vk; R). Example A.4 Let V and U be vector spaces, then the multilinear form F: V ∗ × V × U → R,(ω, v, u) → F(ω, v, u) is a tensor of rank 3. 1 n Example A.5 Let (e1,...,en) be the basis vectors of V and (ε ...,ε ) the corre- sponding dual basis. Then the basis vectors ε j : V → R , v → ε j (v) = v j , ∗ ei : V → R , ω → ei (ω) = ωi , 1 There exists also an abstract definition of tensor spaces as quotient spaces of free vector spaces, cf. [1], Chap.12. Since we are assuming the vector spaces to be finite dimensional, there exists a canonical isomorphism between the abstract tensor space and the space of multilinear forms. Accordingly, the two terms multilinear form and tensor are used synonymously. 126 Appendix A: Multilinear Algebra are tensors of rank 1, projecting the vector v and the covector ω to their j-th and i-th component, respectively. Definition A.6 (Tensor Product)LetV1,...,Vk, W1,...,Wl be vector spaces, and let F ∈ L(V1,...,Vk; R) and G ∈ L(W1,...,Wl ; R). Define a function F ⊗ G: V1 ×···×Vk × W1 ×···×Wl → R by (F ⊗ G)(v1,...,vk, w1, ··· , wk) := F(v1,...,vk)G(w1, ··· , wk). (A.8) From the multilinearity of F and G it follows that (F ⊗ G) is multi linear too. So, F ⊗ G ∈ L(V1,...,Vk, W1,...,Wl ; R) and is called the tensor product of F and G. Example A.7 Let U and V be vector spaces, F: V × U → R and g: V ∗ → R.The tensor product of F and g is given by the multilinear form F ⊗ g: V × U × V ∗ → R , defined by its application on the vectors (v, u, ω) as (v, u, ω) → (F ⊗ g)(v, u, ω) = F(v, u)g(ω). Applying the definition of the tensor product (A.8) several times, it follows straight forward, that the tensor product is bilinear and associative. Insofar it is allowed to write the tensor products between the tensors F, G and H without any brackets as F ⊗ (G ⊗ H) = (F ⊗ G) ⊗ H =: F ⊗ G ⊗ H . With the tensor product, it is possible to find the building blocks of tensors of arbi- trary rank. The following proposition is formulated and proved for L(U, V, V ∗; R), but extends on a basis of the space of multilinear forms. Proposition A.8 Let U, V be vector spaces of dimensions k and l with bases 1 k 1 l (b1,...,bk) and (e1,...,el ), respectively. Let (β ,...,β ) and (ε ,...,ε ) be the corresponding dual bases of U ∗ and V ∗, respectively. Then the set α i B = β ⊗ ε ⊗ e j : 1 ≤ α ≤ k, 1 ≤ i, j ≤ l is a basis for F ∈ L(U, V, V ∗; R), which therefore has dimension kl2. Summing repeated greek indices from 1 to k and repeated roman indices from 1 to l, any multilinear form F can be written as a linear combination Appendix A: Multilinear Algebra 127 = j βα ⊗ εi ⊗ , F Fαi e j (A.9) j = ( , ,εj ) with Fαi F bα ei . Proof The proof is very similar to the proof of Proposition A.2. We need to show that B is linearly independent and spans L(U, V, V ∗; R).Letu ∈ U, v ∈ V and ω ∈ V ∗. Due to the multilinearity of F it follows directly that α i j j α i F(u, v, ω) = F(u bα,v ei ,ωj ε ) = F(bα, ei ,ε )u v ω j . (A.10) Writing the multilinear form as a linear combination (A.9) and applying it to the same vectors u, v and ω, it follows by the definition of the tensor product that ( , , ω) (A=.9) ( j βα ⊗ εi ⊗ )( , , ω) F u v Fαi e j u v (A=.8) j βα( β )εi (vm ) (ω εn) Fαi u bβ em e j n (A.11) (A=.3) j β vmω δα δi δn (A=.4) j αvi ω . Fαi u n β m j Fαi u j j = ( , ,εj ) B Let Fαi F bα ei . Then comparison of (A.10) and (A.11) proves that spans L(U, V, V ∗). To show that B is linearly independent, suppose some linear combination equals zero: = j βα ⊗ εi ⊗ = F Fαi e j 0 Applying the multilinear form F to arbitrary vectors u ∈ U, v ∈ V and ω ∈ V ∗,it follows by the computation of (A.11) that ∗ ∀u ∈ U, v ∈ V, ω ∈ V : F(u, v, ω) = 0 ⇒∀ α,vi ,ω ∈ R: j αvi ω = u j Fαi u j 0 ⇒ j = .
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