What is more general than multilinear algebra

11 Multilinear Algebra

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1 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page Multilinear Algebra 11.1 Multilinear mappings Definition: Consider K-vector spaces V 1, ..., V r and W. A map ϕ: V 1 ... V r W, (v 1, ..., vr) ϕ (v 1, ..., vr), which is separately linear in each variable vi, is called multilinear. We denote the set of all such by Mult K (V 1, ..., V r; W). Proposition: This is a subspace of the space of all mappings V 1 ... V r W. Special case: For r = 1, Mult K (V; W) = Hom K (V, W), compare 5.9. In particular, Mult K (V; K) = Hom K (V, K) = V is the dual space of V, compare special case: For r = 2, multilinear also means bilinear. In particular, Mult K (V, V; K) is the space of all bilinear forms on V, compare 9.3. Proposition: (functoriality) Linear maps fi: V i V i and g: WW induce a linear map Mult K (V 1, ..., V r; W) Mult K (V 1, ..., V r; W ), ϕ g ϕ (f 1 ... fr). Proposition: Consider bases B i of V i and C of W. Consider a system of coefficients αb c 1, ..., br K for all bi B i and c C with the property bi B i: {c C α c b1 , ..., br 0} <. Then there is exactly one multilinear mapping ϕ: V 1 ... V r W, so that for all b i B i the following applies: ϕ (b 1, ..., b r) = c C α c b1, ..., b r c. Conversely, every multilinear mapping ϕ: V 1 ... V r W has this form for unique coefficients α c b 1, ..., b r. Proposition: For any V 1, ..., V r, w, with the convention 0 = 0: (r) dim K Mult K (V 1, ..., V r; W) = dim K (Vi) dim K (W). Consequence: (a) Mult K (V 1, ..., V r; W) 0 if and only if all V i, W are 0. (b) It is Mult K (V 1, ..., V r; W) 0 and finite-dimensional if and only if all V i, W 0 and are finite-dimensional. i = 1

2 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page Symmetrical and alternating mappings Definition: A multilinear map ϕ: V r W is called symmetric, if v 1, ..., vr V σ S r: ϕ (v σ1 ,. .., v σr) = ϕ (v 1, ..., vr). It is called alternating if v 1, ..., v r V: () i i: v i = v i ϕ (v1, ..., v r) = 0. We denote the set of all such by Sym r K (V, W) or Alt r K (V, W). Proposition: These are subspaces of Mult K (V, ..., V; W). Remark: Since every permutation is a product of transpositions of neighboring indices, ϕ is symmetric if and only if v 1, ..., vr V 2 ir: ϕ (v 1, ..., vi 2, vi, vi 1, v i + 1, ..., vr) = ϕ (v 1, ..., vr). Variant: A multilinear mapping ϕ: V r W is called antisymmetric, if v 1, ..., vr V σ S r: ϕ (v σ1, ..., v σr) = sgn (σ) ϕ (v 1, ..., from right). Proposition: (a) It always applies alternately antisymmetric. (b) If 2 0 in K, then antisymmetrically alternating applies. (c) If 2 = 0 in K, then we have antisymmetric symmetric. The term antisymmetric is therefore less important than the others. Proposition: (functoriality) Linear maps f: VV and g: WW induce linear maps Sym r K (V, W) Sym r K (V, W), Alt r K (V, W) Altr K (V, W), ϕ g ϕ (f ... f). Proposition: Consider bases B of V and C of W, as well as a multilinear map ϕ: V r W with coefficients αb c 1, ..., br K as in Then ϕ is symmetric if and only if bi B c C σ S r: α cb σ1, ..., b σr = α cb 1, ..., br, and ϕ is alternating if and only if bi B c C: {σ Sr: α cb σ1, ..., b σr = sgn (σ) αc b 1, ..., br and () ii: bi = bi α c b1, ..., br = 0.

3 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page 119 In the special case r = 2 this means that for every c C the matrix A c: = (α cb 1, b 2) b1, b 2 B is symmetrical, resp. antisymmetric with zero diagonal. Proposition: () dim K Sym r K (V, W) = dimk (V) + r 1 dim K (W), r () dim K Alt r K (V, W) = dimk (V) dim K (W ). r consequence: For all r> dim K (V) we have Alt r K (V, W) = 0. For r = dim K (V) we have dim K Alt r K (V, K) = 1. Example: The determinant induces an alternating multilinear mapping different from zero (K n) n K, (v 1, ..., v n) det ((v 1, ..., v n)). For dimensional reasons, this forms a basis for Alt n K (Kn, K). Theorem: For every endomorphism f of a K-vector space V of dimension n

4 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page Tensor Product Consider two K-vector spaces V 1 and V 2. Definition: A tensor product of V 1 and V 2 over K consists of a K-vector space Ṽ and a bilinear map κ: V 1 V 2 Ṽ with the universal property: For every K-vector space W and every bilinear mapping ϕ: V 1 V 2 W there is exactly one linear mapping ϕ: Ṽ W with ϕ κ = ϕ, that is, so that the following diagram commutes : ϕ V 1 V 2 κ Ṽ Proposition: A tensor product is unique except for unique isomorphism, in other words: If both (Ṽ, κ) and (Ṽ, κ) are tensor products of V 1 and V 2, then there is a unique isomorphism i: Ṽ Ṽ with i κ = κ, that is, so that the following diagram commutes: ϕ κ V 1 V 2 = κ i Ṽ Theorem: A tensor product always exists. Convention: We fix a tensor product (Ṽ, κ) once and for all and denote the vector space Ṽ with V 1 K V 2 or V 1 V 2 for short, as well as the mapping κ with Then we forget the notation (Ṽ, κ). W Ṽ V 1 V 2 V 1 K V 2, (v 1, v 2) v 1 v 2. Remark: So we don't care how the tensor product is constructed, but just use its universal property. The uniqueness except for a clear (!) Isomorphism has the effect that each element of a second choice of (Ṽ, κ) corresponds to a unique element of the first choice, and that these elements each fulfill the same formulas and have the same other properties. Calculation rules: The bilinearity of κ translates into the following calculation rules for all vi, vi V i and λ K: (v 1 + v 1) v 2 = v 1 v 2 + v 1 v 2 λv 1 v 2 = λ (v 1 v 2) v 1 (v 2 + v 2) = v 1 v 2 + v 1 v 2 v 1 λv 2 = λ (v 1 v 2)

5 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page 121 Proposition: (adjunction formula) There are unambiguous isomorphisms Hom K (V 1 KV 2, W) = Mult K (V 1 V 2, W) = Hom K (V 1, Hom K (V 2, W)) with f ϕ f (v 1 v 2) = ϕ (v 1, v 2) = ψ (v 1) (v 2). ψ Proposition: (functoriality) For linear maps fi: V i V i a linear map exists exactly f 1 f 2: V 1 KV 2 V 1 KV 2 with v 1 v 2 f 1 (v 1) f 2 (v 2) . fi Proposition: For all linear mappings V i V gii V i applies (a) id V1 id V2 = id V1 V 2. (b) f 1 0 V2 = 0 V1 f 2 = 0 V1 V 2. (c) (g 1 g 2) (f 1 f 2) = (g 1 f 1) (g 2 f 2). (d) If f i is an isomorphism with inverse g i, then f 1 f 2 is an isomorphism with inverse g 1 g 2. Theorem: Let B i be a basis of V i. Then the b 1 b 2 are different for all (b 1, b 2) B 1 B 2, and {b 1 b 2 (b1, b 2) B 1 B 2} is a basis of V1 KV 2. In particular, dim K (V 1 KV 2) = dim K (V 1) dim K (V 2). Example: For all natural numbers m, n there is a natural isomorphism K m KK n Mat mn (K) with vwvw T. Proposition: For all V and W there is a natural injective homomorphism VKW Hom K (V, W) with lw (vl (v) w). Its image is the subspace of all homomorphisms of finite rank. In particular, it is an isomorphism if and only if V or W is finite-dimensional. Definition: An element of V 1 K V 2 is called a tensor. An element of the form v 1 v 2 is called a pure tensor. Proposition: The pure tensors generate V 1 K V 2. Proposition: Consider vectors v i, v i V i. (a) It is v 1 v 2 0 exactly if v 1, v 2 0. (b) In case (a), v 1 v 2 = v 1 v 2 if and only if λ K: (v 1, v 2) = (λv 1, λ 1 v 2). (c) If v i, v i are linearly independent, then v 1 v 2 + v 1 v 2 is not a pure tensor.

6 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page Higher tensor products Proposition: There are unique isomorphisms, characterized as follows: V 1 KKV 1 with v 1 1 v 1 (identity) V 1 KV 2 V 2 KV 1 with v 1 v 2 v 2 v 1 (commutativity) (V 1 V 2) V 3 V 1 (V 2 V 3) with (v 1 v 2) v 3 v 1 (v 2 v 3) (associativity) define any finite sequence of vector spaces V 1 K ... KV r without brackets. This carries a natural multilinear mapping κ: V 1 ... V r V 1 K ... K V r, (v 1, ..., v r) v 1 ... v r. Together they have the universal property: For every K-vector space W and every multilinear mapping ϕ: V 1 ... V r W there is exactly one linear mapping ϕ: V 1 K ... KV r W with ϕ κ = ϕ, that means that the following diagram commutes: V 1 ... V r ϕ W κ V 1 K ... KV r ϕ Proposition: dim K (V 1 K ... KV r) = r dim K (V i) . i = 1 Definition: For every natural number r the r-th tensor power of V is defined by V 0: = K or V r: = V K ... KV with r factors for r 1. Definition: The space T r, s (V): = V r K (V) s is called the space of the r-fold covariant and s-fold contravariant tensors, or short-term tensors of the type (r, s). Another convention calls them of type (s, r). Remark: If dim K (V) <, then every basis of V yields the corresponding dual basis of V and thus a basis of T r, s (V). In particular, dim K T r, s (V) = (dim K V) r + s. Remark: Depending on the situation, an n n-matrix can represent a tensor of the type (2, 0) or (1, 1) or (0, 2) for the space V = K n. For each of these types the base change with the Base change matrix U described by another formula, namely by AUT AU or U 1 AU or U 1 A (UT) 1. The best way to avoid this confusion is to stick to the abstract terms where V and V as long as possible can be clearly distinguished by the notation.

7 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page 123 Proposition: For every one-dimensional K-vector space V there is a natural isomorphism V K V K with l v l (v). Example: Every scalar physical quantity lies in a certain one-dimensional R-vector space, and the choice of a basic unit corresponds to the choice of a basic vector. Only physical quantities of the same kind, i.e. elements of the same vector space, can be compared with one another. Examples: size vector space basic unit relation time T second, hour h = 3600s length L meter, inch in = 0.0254m mass M kilogram, pound lb = kg compound scalar physical quantities naturally lie in certain tensor spaces. Examples: scalar quantity vector space basic unit area L 2 m 2 frequency T 1 / s speed LT m / s acceleration L (T) 2 m / s 2 force ML (T) 2 N = kg m / s 2 energy ML 2 (T) 2 J = kg m 2 / s 2 Classical vector physical quantities lie in Euclidean vector spaces. For example, let R be the three-dimensional space. Some other quantities are then: vector quantity speed acceleration impulse force voltage vector space RTR (T) 2 MRTMR (T) 2 ML (T) 2 (R) 2 Compare also the contribution of Terence Tao: Quantum mechanical physical states lie in unitary vector spaces, which often have a less direct connection with the classic local space.

8 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page Symmetric and alternating powers Definition: For every natural number r there is an r-th symmetric, or alternating, power of V over K from a K-vector space Ṽ and a symmetric, resp alternating, multilinear mapping κ: V r Ṽ with the universal property: For every K-vector space W and every symmetrical or alternating, multilinear mapping ϕ: V r W there is exactly one linear mapping ϕ: Ṽ W with ϕ κ = ϕ , that is, so that the following diagram commutes: V r ϕ κ Ṽ ϕ W Proposition: A symmetric, or alternating, power is unambiguous except for unambiguous isomorphism, in other words: If both (Ṽ, κ) and (Ṽ, κ) such a power of V, there is a clear isomorphism i: Ṽ Ṽ with i κ = κ, that is, so that the following diagram commutes: V r κ Ṽ = κ i Ṽ proposition: A symmetrical or alternating, Potency always exists. Convention: We fix once and for all an r-th symmetric power and denote the associated vector space with S r KV or Sr V, as well as the associated symmetrical multilinear mapping with V r S r KV, (v 1, ..., vr) v 1 v r. We fix once and for all an r-th alternating power and denote the associated vector space with Λ r KV or Λr V, as well as the associated alternating multilinear mapping with V r Λ r KV, (v 1, ..., vr) v 1 ... v r. Calculation rules: The multilinearity and symmetry properties of these figures mean certain calculation rules. In the case r = 2 these are for all v, w, w V and λ K: v (w + w) = vw + vwv λw = λ (vw) wv = vwv (w + w) = vw + vwv λw = λ (vw) vv = 0 wv = vw

9 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page 125 Proposition: (adjunction formula) There are unambiguous isomorphisms Hom K (S r KV, W) Hom K (Λ r KV, W) Sym r K (V, W), f ((v 1, ..., vr) f (v 1 vr)), Alt r K (V, W), f ((v 1, ..., vr) f (v 1 ... vr)) . Proposition: (functoriality) For every linear mapping f: VV there are unique linear maps S rf: S r KV Sr KV with v 1 vrf 1 (v 1) fr (vr), Λ rf: Λ r KV Λr KV with v 1. .. vrf 1 (v 1) ... fr (vr). Proposition: For all linear mappings V (a) S r id V = id S r V and Λ r id V = id Λ r V. (b) If f = 0, then S r f = 0 and r f = 0. (c) S r g S r f = S r (g f) and λ r g Λ r f = Λ r (g f). f V g V applies (d) If f is an isomorphism with inverse g, then S r f or Λ r f is an isomorphism with inverse S r g or Λ r g. Special case: V 0 = S 0 V = Λ 0 V = K and V 1 = S 1 V = Λ 1 V = V. Theorem: Let B be a basis of V, provided with a total order or .. Then the elements form b 1 br for all bi B with b 1 ... br a basis of S r KV, respectively. b 1 ... br for all bi B with b 1 ... br a basis of Λ r K V. In particular, () dim K SK r V = dimk (V) + r 1, r () dim K Λ r KV = dimk (V). r consequence: For all r> dim K (V) we have Λ r K V = 0. For r = dim K (V) we have dim K Λ r K V = 1. Theorem: For every endomorphism f of a K-vector space V of dimension n

10 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page Tensalgebra, symmetric, outer algebra Proposition: For all r, s 0 there are unique bilinear maps: V r V s V (r + s) with (v 1 ... vr) (v r + 1 ... v r + s) = v 1 ... v r + s,: S r VS s VS r + s V with (v 1 vr) (v r + 1 v r + s) = v 1 v r + s,: Λ r V Λ s V Λ r + s V with (v 1 ... vr) (v r + 1 ... v r + s) = v 1 ... v r + s. Definition: (For the outer direct sum see 5.4.) Tensor algebra TV: = r0 V r (ξ r) r0 (η s) s0: = (tr = 0 ξ r η tr) t0 symmetric algebra SV: = r0 S r V (ξ r) r0 (η s) s0: = (tr = 0 ξ r η tr) t0 outer algebra ΛV: = r0 Λ r V (ξ r) r0 (η s) s0: = (tr = 0 ξ r η tr) t0 Proposition: With the addition of the underlying vector space and the given multiplication as well as the unit element of K = V 0 = S 0 V = Λ 0 V, this is an associative unitary graduated ring. The calculation rules applicable in the respective ring result from those in 11.3 and 11.5 as well as the definition above. Special case: For dim K V = 0 we have V r = S r V = Λ r V = 0 for all r> 0, and therefore TV = SV = ΛV = K. Special case: Let dim K V = 1 with basis b. For all r 0 then dim K (V r) = 1 with base br and dim K (S r V) = 1 with base br, and there are unique isomorphisms K [X] K Kb TV SV, ΛV with bb = 0 . i0 a ix i i0 a ib i i0 a ib i, or proposition: (a) For V 0, dim K (TV) = dim K (SV) =. (b) For dim K (V) <, dim K (ΛV) = 2 dim K (V), otherwise dim K (ΛV) =. Proposition: (a) The ring SV is always commutative. (b) For dim K V 1 TV and ΛV are commutative. (c) If 2 = 0 in K, then ΛV is commutative. (d) For dim K V 2, TV is not commutative. (e) For dim K V 2and2 0in K, ΛV is not commutative.

11 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page 127 Proposition: For all r, s 0 and all ξ Λ r V and η Λ s V we have η ξ = (1) rs ξ η. In particular, for even r, every element of Λ r V commutes with all of ΛV. Proposition: If dim K (V) <, then for all r 0natural isomorphisms exist Λ r (V) ϕ r Alt r (V, K) (Λ r V). The second is a special case of the adjunct formula, and the first arises from the multilinear form (V) r V r K, (l, which alternates separately in (l 1, ..., lr) and (v 1, ..., vr) 1, ..., lr, v 1, ..., vr) σ S r sgn (σ) l 1 (v σ1) lr (v σr). Proposition: For all r, s 0 the following diagram commutes Λ r (V) Λ s (V) Λ r + s (V) ϕ r ϕ s Alt r (V, K) Alt s (V, K) ϕ r + s Alt r + s (V, K) where the figure below is defined by the formula (ϕ ψ) (v 1, ..., v r + s): = σ sgn (σ) ϕ (v σ1, .. ., v σr) ψ (v σ (r + 1), ..., v σ (r + s)) and the sum extends over all σ S r with σ1 <... <σrund σ (r +1) <... <σ (r + s) extends.Note: In the course Analysis II, differential forms were introduced as alternating multilinear forms and their outer product by the above alternating sum. This somewhat artificial-looking formula becomes much simpler and more natural when applied to the external power Λ r (V).

12 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page Vector product in R 3 For every K-vector space V of dimension n 0. (g) vw is the area of ​​the parallelogram spanned by v and w, i.e. vw = vw sin ϑ if v, w = vw cos ϑ.

13 11 Multilinear Algebra Pink: Lineare Algebra 2014/15 page Body extension Let K be a sub-body of a body L, that is, a subset which itself forms a body with the arithmetic operations induced by L. Then every L-vector space with the same addition and the scalar multiplication restricted to K is also a K-vector space. In particular, L itself becomes a K-vector space. Proposition: For every K-vector space V there is exactly one structure as an L-vector space on V K L, whose additive group is that of V K L and for which the following applies: x, y L v V: x (v y) = v xy. Definition: The L-vector space VL with respect to K L.: = VKL is called the basis extension of V Proposition: For every basis B of the K-vector space V, the elements b 1of VL form a basis BL of the L-vector space V L for all b B. In particular, dim L (VL) = dim K (V). Example: For every n 0 there exists a natural isomorphism of L-vector spaces K n K L L n with v x xv. Definition: In the case of RC, VC means the complexification of the real vector space V. Definition: The complex conjugate of a C vector space (W, + ,, 0 W) is the C vector space (W, + ,, 0 W), in which the scalar multiplication has been replaced by: CWW, (z, w) zw: = z w. As we usually abbreviate (W, + ,, 0 W) with W, we write for (W, + ,, 0 W) only briefly W. Example: For every C-subspace WC n there is a natural isomorphism of C-vector spaces W {ww W} C n, w w. Remark: For every C-vector space W we have (W) = W. Remark: Every basis of W is also a basis of W. Proposition: (Compare 9.11 and 10.7.) For every finite-dimensional unitary vector space W there is a natural isomorphism of C-vector spaces δ: WW: = Hom C (W, C) , v δ (v): = v ,. Proposition: For every C-vector space W there is a natural isomorphism of C-vector spaces W R C W W, w z (zw, z w).