Multilinear algebra

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$Math 126 Solution Set 2$

$Math S?101. Worksheet 1. Axioms of Integer Arithmetic$

Exercise 2.5. Give a definition of a trilinear map U × V × W → E (it should be something like “linear in all three variables simultaneously”). Show that U ⊗ V ⊗ W (parenthesized in either way by Exercise 2.2) is the universal vector space with a trilinear map from U × V × W (the same way V ⊗ W is the universal vector space with a bilinear map from V × W ). Exercise 2.6. Generalize Exercise 2.5 by replacing 3 by n: define an n-multilinear map V1×V2×· · ·×Vn → E, and show that V1 ⊗V2 ⊗· · ·⊗Vn is the universal vector space with an n-multilinear map V1 ×V2 ×· · ·×Vn → V1 ⊗ V2 ⊗ · · · ⊗ Vn. Exercise 2.7. Define a product map of abelian groups to be a map µ : A × B → C satisfying µ(a + a ′ , b) = µ(a, b) + µ(a ′ , b) and µ(a, b + b ′ ) = µ(a, b) + µ(a, b ′ ). (a): Show that for any two abelian groups A and B, a universal product A × B → A ⊗ B exists. (Hint: Imitate the proof of Theorem 2.1.) (b): Show that Z/2 ⊗ Z/3 = 0. (Hint: Show that any product Z/2 × Z/3 → C is identically 0.) 6
3 Exterior products and determinants In this section, we will look at a specific notion of multiplication, namely one that relates to area and volume. The basic idea is as follows. Given two vectors v and w, we can form the parallelogram that they span, and write v ∧ w for something we think of as the “area” of the parallelogram. This is not quite the usual notion of area, however, because we want to think of it not just as a single number but as also having a “two-dimensional direction” (the same way a single vector v both has a size and a direction). That is, if we had a parallelogram pointing in a “different direction”, i.e. in a different plane, we would think of it as different. Let’s write down some properties this v ∧ w ought to have. Scaling v or w scales the parallelogram, and thus ought to scale its area. That is, for scalars c, we should have (cv) ∧ w = c(v ∧ w) = v ∧ (cw). This suggests that the operation ∧ ought to be bilinear. Another property of ∧ is that for any vector v, v ∧ v should be 0: if both vectors are pointing in the same direction, the “parallelogram” is just a line segment and has no area. As it turns out, these two properties are the only properties we really need. Definition 3.1. Let V be a vector space. Then the exterior square 2 (V ) is the quotient of V ⊗ V by the subspace spanned by the elements v ⊗ v for all v ∈ V . We write v ∧ w for the image of v ⊗ w under the quotient map V ⊗ V → 2 (V ). Let’s try to understand what this vector space 2 (V ) actually looks like. First, there is a very important formal consequence of the fact that v ∧ v = 0 for all v. Namely, if we apply this to a sum v + w, we get 0 = (v + w) ∧ (v + w) = v ∧ v + v ∧ w + w ∧ v + w ∧ w = v ∧ w + w ∧ v. Thus for any v and w, v ∧ w = −w ∧ v. That is, as a “multiplication”, this ∧ operation is “anticommutative” (or “alternating”). We can now see what 2 (V ) looks like if we pick a basis {ei} for V . We know that {ei ⊗ ej} is a basis for V ⊗ V . However, in 2 (V ), ei ∧ ei = 0 and ei ∧ ej = −ej ∧ ei, so 2 (V ) can be spanned by just the elements ei ∧ ej for which i < j. Seeing that all these are indeed linearly independent is a bit trickier. Theorem 3.2. Suppose {ei} is a basis for V . Then {ei ∧ ej}i j, and T (ei ⊗ ei) = 0. We want to show that T gives a map S : 2 (V ) → E; it suffices to show that T (v ⊗ v) = 0 for all v ∈ V . Let v = ciei; then v ⊗ v = cicjei ⊗ ej = c 2 i ei ⊗ ei + cicj(ei ⊗ ej + ej ⊗ ei). i We thus see that T (v ⊗ v) = 0. Hence T gives a map S : 2 (V ) → E which sends ei ∧ ej to eij. Since the eij are linearly independent in E by construction, this implies that the ei ∧ ej (for i < j) are linearly independent, and hence a basis. One thing to note about 2 (V ) is that not every element is of the form v ∧ w. That is, not every “area vector” is just an area in some plane; it can also be a sum of areas in different planes. For example, if {ei} is a basis, then e1 ∧ e2 + e3 ∧ e4 cannot be simplified to a single v ∧ w. We can use the same idea to generalize from area to volume, or even n-dimensional volume. 7 i
Page 1 and 2: 1 Universal sums and products Multi
Page 3 and 4: (a): Show that Hom(V, W ) is natura
Page 5: Proof. First, we define a map T : U
Page 9 and 10: 4 More on Determinants; Duality Let
Page 11 and 12: Exercise 4.3. Define a map T : V
Page 13: That is, a V on one side of a linea

Multilinear algebra

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