STEVEN DALE CUTKOSKY

1. Products Theorem 1.1. Suppose that V , W are finite dimensional vector spaces over a field F . Then there exists a T over F , and a ϕ : V × W → T such that T satisfies the following “universal property”: If U is a vector space over F and g : V × W → U is a bilinear map then there exists a unique g∗ : T → U such that for all (v, w) ∈ V × W we have g(v, w) = g∗(ϕ(v, w)).

Lemma 1.2. Suppose that T1, with bilinear map ϕ1 : V × W → T1, satisfies the universal property of theorem 1.1, and T2, with bilinear map ϕ2 : V ×W → T2, satisfies the universal property of theorem 1.1. Then there exists a (unique) linear isomorphism ψ : T1 → T2 such that ψϕ1 = ϕ2. Thus the vector space T of the conclusions of Theorem 1.1 is uniquely determined up to isomorphism by the universal property. We denote T by V N W and call T the “ of V and W over F ”. We denote ϕ(v, w) = v ⊗ w for v ∈ V and w ∈ W , so that g(v, w) = g∗(v ⊗ w). Since ϕ : V × W is bilinear,we have that (x + x0) ⊗ y = (x ⊗ y) + (x0 ⊗ y), (cx) ⊗ y = c(x ⊗ y), (1) x ⊗ (y + y0) = (x ⊗ y) + (x ⊗ y0), x ⊗ (cy) = c(x ⊗ y) for x, x0 ∈ V , y, y0 ∈ W and c ∈ F . Theorem 1.1 states that there is a “commutative diagram” ϕ V × W → V N W g & ↓ g∗ . U

That is, g∗ϕ = g. To prove Theorem 1.1, we will use the following theorem. Theorem 1.3. Let S be a set of objects, and F be a field. Then there exists a vector space V (S) over F which contains S, and S is a of V (S) over F .

The elements of V (S) are “formal” linear combinations c1v1+···+cnvn with v1, . . . , vn ∈ S and c1, . . . , cn ∈ F . Two sums c1v1 + ··· + cnvn and d1v1 + ··· + dnvn in V (S) are equal precisely when ci = di for 1 ≤ i ≤ n.

We now prove Theorem 1.1. Let {v1, . . . , vm} be a basis of V and let {w1, . . . , wn} be a basis of W . Let ti,j be a symbol for 1 ≤ i ≤ m and 1 ≤ j ≤ n. Let T = V ({ti,j | 1 ≤ i ≤ m and 1 ≤ j ≤ n}) be the vector space over F consisting of formal linear combinations of 1 the ti,j, so that {ti,j | 1 ≤ i ≤ m and 1 ≤ j ≤ n} is a basis of T . The elements of T are linear combinations m n X X cijti,j i=1 j=1 with cij ∈ F . Define a bilinear map ϕ : V × W → T by m n X X ϕ(v, w) = xiyjti,j i=1 j=1 Pm Pn for v = i=1 xivi ∈ V and w = j=1 yjwj ∈ W . We now prove that T with the bilinear map ϕ has the desired universal property. Suppose that g : V × W → U is a bilinear map. Since {tij} is a basis of T , there exists a unique linear map g∗ : T → U such that g∗(tij) = g(vi, wj) for 1 ≤ i ≤ m and 1 ≤ j ≤ n. Since ϕ(vi, wj) = tij, we see that the map g∗ of the conclusions of the Theorem exists and is uniquely determined.

Proof of Lemma 1.2: Since T1 with bilinear map ϕ1 : V × W → T1 satisfies the universal property, we have a unique linear map (ϕ2)∗ : T1 → T2 such that (ϕ2)∗ϕ1 = ϕ2. These maps form a commutative diagram

ϕ1 V × W → T1 (2) ϕ2 & ↓ (ϕ2)∗ . T2

Since T2 with bilinear map ϕ2 : V × W → T2 satisfies the universal property, we have a unique linear map (ϕ1)∗ : T2 → T1 such that (ϕ1)∗ϕ2 = ϕ1. These maps form a commutative diagram

ϕ2 V × W → T2 (3) ϕ1 & ↓ (ϕ1)∗ . T1

Thus (ϕ1)∗(ϕ2)∗ : T1 → T1 satisfies (ϕ1)∗(ϕ2)∗ϕ1 = ϕ1. That is, combining the commuta- tive diagrams (2) and (3), we obtain a commutative diagram

ϕ1 V × W → T1 (4) ϕ1 & ↓ (ϕ1)∗(ϕ2)∗ . T1

Now we have that the identity map IT1 : T1 → T1 satisfies the condition that IT1 ϕ1 = ϕ1, so by uniqueness in the universal property of T1, we have that IT1 = (ϕ1)∗(ϕ2)∗. Combining (3) and (2), and applying the the universal property of T2, we obtain that the identity map of T2 is IT2 = (ϕ2)∗(ϕ1)∗. Thus (ϕ1)∗ is an isomorphism of T1 and T2 with inverse (ϕ2)∗.

Lemma 1.4. If V and W are finite dimensional vector spaces, of respective dimensions m and n, then V ⊗ W has dimension mn.

Proof. This follows from the construction of T in the proof of Theorem 1.1. 

Lemma 1.5. Suppose that {αi | 1 ≤ i ≤ m} is a basis of V and {βj | 1 ≤ j ≤ n} is a basis of W . Then {αi ⊗ βj | 1 ≤ i ≤ m and 1 ≤ j ≤ n} is a basis of V ⊗ W . 2 Proof. One proof of this follows from our construction of V ⊗ W , which is applicable for any bases of V and W (and the uniqueness of this construction up to isomorphism by Lemma 1.2). A direct proof is obtained as follows. Let {v1, . . . , vm} be the basis of V and {w1, . . . , wn} be the basis of V ⊗ W used in our construction of V ⊗ W , so that {vi ⊗ wj | 1 ≤ i ≤ m and 1 ≤ j ≤ n} is a basis of V ⊗ W . Expand m X vi = xikαk k=1 and n X wj = yjlβl. l=1 We obtain m n X X vi ⊗ wj = xikyjkαk ⊗ βl k=1 l=1 for 1 ≤ i ≤ m and 1 ≤ j ≤ n. Thus the mn vectors {αi ⊗ βj} span V ⊗ W , as {vi ⊗ wj} is a basis of V ⊗ W . Since the mn vectors {vi ⊗ wj} are a basis of V , we have that V ⊗ W has dimension mn, so that {αi ⊗ βj} is a basis of V ⊗ W . 

Warning: Although the vector space V ⊗ W is spanned by elements of the form v ⊗ w with v ∈ V , w ∈ W , not every element of V ⊗ W can be written in this form (unless one of V , W has dimension 1).

The following Proposition follows from the construction of Theorem 1.1 and from Lemma 1.2. Lemma 1.6. Suppose that V and W are finite dimensional vector spaces. Then there is a unique isomorphism V ⊗ W → W ⊗ V such that x ⊗ y 7→ y ⊗ x for x ⊗ y ∈ V ⊗ W . Proof. The map V × W → W ⊗ V such that (x, y) 7→ y ⊗ x is bilinear, and thus factors through the tensor product V ⊗ W , sending x ⊗ y to y ⊗ x. Since this last map has an inverse (by symmetry) we obtain the desired isomorphism. 

Warning: If v ∈ V and w ∈ V are distinct vectors in a vector space V , then v⊗w 6= w⊗v in V ⊗V . The above Lemma only says that the map v⊗w 7→ w⊗v is a linear isomorphism of V ⊗ V , which is not the identity. Lemma 1.7. Suppose that U, V and W are finite dimensional vector spaces. Then there is a unique isomorphism (U ⊗ V ) ⊗ W → U ⊗ (V ⊗ W ), such that (x ⊗ y) ⊗ z 7→ x ⊗ (y ⊗ z) for x ∈ U, y ∈ V and z ∈ W .

Proof. Let {ui} be a basis of U, {vj} be a basis of V , and {wk} be a basis of W . Then the elements (ui ⊗ vj) ⊗ wk form a basis of (U ⊗ V ) ⊗ W , and the elements ui ⊗ (vj ⊗ wk) form a basis of U ⊗ (V ⊗ W ). By the general theorem on the existence and uniqueness of linear maps, there exists a unique linear map F :(U ⊗ V ) ⊗ W → U ⊗ (V ⊗ W ) 3 which maps (ui ⊗ vj) ⊗ wk on ui ⊗ (vj ⊗ wk). Now by taking linear expansions in terms of the basis, we see that F ((u ⊗ v) ⊗ w) = u ⊗ (v ⊗ w) for u ∈ U, v ∈ V , w ∈ W . It follows that F has the desired effect. Since F maps a basis of (U ⊗ V ) ⊗ W on a basis of U ⊗ (V ⊗ W ), it follows that F is an isomorphism. 

This Lemma tells us that we can unambiguously write U1 ⊗ U2 ⊗ · · · ⊗ Un for the tensor product of vector spaces U1,...,Un, and u1 ⊗· · ·⊗un for the the tensor product of elements of U ,...,U . If {u1 }, ··· , {un } are any bases of U ,...,U , then {u1 ⊗ · · · ⊗ un } is a 1 n i1 in 1 n i1 in basis of U1 ⊗ · · · ⊗ Un. This product is universal for multilinear maps, analogous to the way that U1 ⊗ U2 is universal for bilinear maps.

Theorem 1.8. Suppose that V1, ··· ,Vn are finite dimensional vector spaces over a field F . Then there exists a vector space T over F , and a multilinear map ϕ : V1 ×V2 ×· · ·×Vn → T such that T satisfies the following “universal property”: If U is a vector space over F and g : V1 × V2 · · · × Vn → U is a multilinear map then there exists a unique linear map g∗ : T → U such that for all (v1, . . . , vn) ∈ V1 × · · · × Vn we have g(v1, ··· , vn) = g∗(ϕ(v1, ··· vn)).

T = V1 ⊗ · · · ⊗ Vn, as defined above, satisfies the universal property of Theorem 1.8.

Lemma 1.9. Suppose that T1, with multilinear map ϕ1 : V1 × · · · × Vn → T1, satisfies the universal property of theorem 1.1, and T2, with multilinear map ϕ2 : V1 × · · · × Vn → T2, satisfies the universal property of Theorem 1.1. Then there exists a (unique) linear isomorphism ψ : T1 → T2 such that ψϕ1 = ϕ2. Lemma 1.10. Let V be a finite dimensional vector space over F . Let V ∗ be the dual space, and L(V,V ) the space of linear maps of V into itself. There exists a unique isomorphism V ∗ ⊗ V → L(V,V ), ∗ which to each element ϕ ⊗ v (with ϕ ∈ V and v ∈ V ) associates the linear map Lϕ⊗v such that Lϕ,v(w) = ϕ(w)v for w ∈ V . ∗ Proof. To each pair (ϕ, v) ∈ V × V associate the linear map Lϕ,v such that

Lϕ,v(w) = ϕ(w)v ∗ for w ∈ V . This association (ϕ, v) 7→ Lϕ,v is a bilinear map V × V → L(V,V ). Thus, by Theorem 1.1, there exists a unique linear map V ∗ ⊗V → L(V,V ) which to ϕ⊗v associates our linear map Lϕ,v. We must now prove that this map

ϕ ⊗ v 7→ Lϕ,v ∗ gives an isomorphism of V ⊗ V and L(V,V ). Let {v1, . . . , vn} be a basis of V , and let ∗ ∗ ∗ {v1, . . . , vn} be the dual basis. Then vi (vk) = δik is the Kronecker delta. Let L = L ∗ ij vi ,vj to simplify notation. We will show that the Lij for 1 ≤ i ≤ n and 1 ≤ j ≤ n are linearly independent. Suppose that we have a relation X X cijLij = 0 i j 4 with cij ∈ F . Apply the left hand side to a vk. We obtain X X 0 = cijLij(vk). j i

In the sum, Lij(vk) = 0 unless i = k, in which case it is equal to vj. Hence X 0 = ckjvj. j

From the linear dependence of v1, . . . , vn, we conclude that ckj = 0 for all j and k, proving 2 that the linear maps Lij are linearly independent. There are n such maps, which is the dimension of L(V,V ). Hence these maps form a basis of L(V,V ). Since our map ∗ ∗ V ⊗ V → L(V,V ) sends a basis {vi ⊗ vj} onto the basis {Lij} of L(V,V ), it follows that this map is an isomorphism.  Lemma 1.11. Suppose that V,W are finite dimensional vector spaces over F . Then there is a unique isomorphism V ∗ ⊗ W ∗ → (V ⊗ W )∗ which to each ϕ ⊗ ψ (with ϕ ∈ V ∗ and ∗ ψ ∈ W ) associates the functional Lϕ,ψ having the property that

Lϕ,ψ(v ⊗ w) = ϕ(v)ψ(w) for v ∈ V and w ∈ W . By the universal property, Theorem 1.1, we may identify (V ⊗V )∗ with the vector space of bilinear forms on V (the bilinear forms are the bilinear maps from V × V to F ). The proceeding Lemma thus allows us to identify V ∗ ⊗ V ∗ with the bilinear forms on a finite dimensional vector space V . We have a multilinear analogue of the above.

Lemma 1.12. Suppose that V1,...,Vn are finite dimensional vector spaces over F . Then ∗ ∗ ∗ there is a unique isomorphism V1 ⊗· · ·⊗Vn → (V1 ⊗· · ·⊗Vn) which to each ϕ1 ⊗· · ·⊗ϕn ∗ (with ϕi ∈ Vi ) associates a functional Lϕ1,··· ,ϕn having the property that

Lϕ1,··· ,ϕn (v1 ⊗ · · · ⊗ vn) = ϕ1(v1) ··· ϕn(vn). By the universal property, Theorem 1.8, we may thus identify (V ∗)⊗n = V ∗ ⊗ · · · ⊗ V ∗ with the vector space of multilinear forms on a finite dimensional vector space V .

Theorem 1.13. Suppose that L1 : U1 → V1 and L2 : U2 → V2 are linear maps. Then there is a unique linear map L1 ⊗L2 : U1 ⊗U2 → V1 ⊗V2 such that u1 ⊗u2 7→ L1(u1)⊗L2(u2) for u1 ∈ U1 and u2 ∈ U2. This map is functorial; that is if M1 : V1 → W1 and M2 : V2 → W2 are linear, then (M1 ⊗ M2) ◦ (L1 ⊗ L2) = (M1 ◦ L1) ⊗ (M2 ◦ L2) are the same maps from U1 ⊗ U2 to W1 ⊗ W2. In particular, if L1 and L2 are isomorphisms, with inverses M1 and M2, then L1 ⊗ L2 is an isomorphism, with inverse M1 ⊗ M2.

L1 and L2 induce a bilinear map L1 × L2 : U1 × U2 → V1 ⊗ V2, which induces a unique linear map L1 ⊗ L2 : U1 ⊗ U2 → V1 ⊗ V2 by the universal property Theorem 1.1, giving a commutative diagram U1 × U2 → U1 ⊗ U2 L1 × L2 ↓ ↓ L1 ⊗ L2 V1 × V2 → V1 ⊗ V2

By the universal property 1.1, L1 ⊗L2 is the unique linear map making the above diagram commutative. 5 Theorem 1.14. Suppose that Li : Ui → Vi are linear maps for 1 ≤ i ≤ n. Then there is a unique linear map L1 ⊗ · · · ⊗ Ln : U1 ⊗ · · · ⊗ Un → V1 ⊗ · · · ⊗ Vn such that u1 ⊗ · · · ⊗ un 7→ L1(u1) ⊗ · · · ⊗ Ln(un) for u1 ∈ U1, . . . , un ∈ Un. This map is functorial; that is if Mi : Vi → Wi are linear for 1 ≤ i ≤ n, then (M1 ⊗ · · · ⊗ Mn) ◦ (L1 ⊗ · · · ⊗ Ln) = (M1 ◦ L1) ⊗ · · · ⊗ (Mn ◦ Ln) are the same maps from U1 ⊗ · · · ⊗ Un to W1 ⊗ · · · ⊗ Wn. In particular, if Li are isomorphisms, with inverses Mi, then L1 ⊗· · ·⊗Ln is an isomorphism, with inverse M1 ⊗ · · · ⊗ Mn.

2. Alternating Products Let V be a vector space, and U be a vector space, over a field F . Let V (r) denote the product of V with itself r times. A multilinear map f : V (r) → U is alternating if f(w1, w2, . . . , wr) = 0 whenever two adjacent entries are equal; that is, whenever there exists an index j < r such that wj = wj+1. An example of an alternating multilinear map is the , Det : F n → F , viewed as a of the columns of a matrix. Suppose that f(w1, . . . , wr) is an alternating multilinear map. As in the proof for , we see that if two adjacent components wj and wj+1 are interchanged, then f changes signs. If two distinct components wi and wj are equal, then f(w1, . . . , wr) = 0. We now introduce some more notation. Let A = (aij) be an r×n matrix, with 1 ≤ r ≤ n. Let S be a subset of the integers {1, . . . , n}, consisting of r elements. The elements of such a set can be ordered, so that if i1, . . . , ir are these elements, then i1 < ··· < ir. Let σ : {1, . . . , r} → S be a 1-1 map. We may then view σ as a permutation of S. If i1, . . . , ir are the elements of S, and are ordered so that i1 < ··· < ir, then σ gives rise to the permutation i1 7→ σ(1), i2 7→ σ(2), . . . , ir 7→ σ(r).

The sign of this permutation is denoted by εS(σ). Let P (S) denote the set of 1-1 maps σ : {1, . . . , r} → S, which we have seen can be identified with the permutations of S. Let A = (aij) be an r ×n matrix, with r ≤ n. For each subset S of {1, . . . , n} consisting of precisely r elements, we take the r × r submatrix of A consisting of those elements aij such that j ∈ S. We denote by DetS(A) the determinant of this submatrix. We have that X DetS(A) = εS(σ)a1,σ(1) ··· ar,σ(r). σ∈P (S)

Now let v1, . . . , vr be elements of V . For each S as above, we let

vS = (vi1 , . . . , vir ), where i1, . . . , ir are elements of S so ordered that i1 < ··· < ir. Theorem 2.1. Let V , U be vector spaces over F . Let f : V (r) → U be an r-multilinear map. Let v1, . . . , vr be elements of V , and let A = (aij) be an r × n matrix with coefficients in F . Let

u1 = a11v1 + ··· + a1nvn . . ur = ar1v1 + ··· + arnvn. 6 Then X f(u1, . . . , ur) = DetS(A)f(vS) S where the sum is taken over all subsets S of {1, . . . , n} consisting of precisely r elements. Proof. If r > n the only alternating multilinear map f : V (r) → U to a vector space U is the zero map (for any w1, . . . , wr ∈ V , with r > n, one of the wi must be expressable as a linear combination of the others. Thus f(w1, . . . , wr) = 0). Further we have that there are no subsets S of {1, . . . , n} consisting of r > n elements. We may interpret the empty sum in the formula as being 0, to obtain the (trivial) validity of the Theorem in the case when r > n. Now assume that r ≤ n. We have

f(u1, . . . , ur) = f(a11v1 + ··· + a1nvn, . . . , ar1v1 + ··· + arnvn). Expanding out by multilinearity, we obtain a sum X a1,σ(1) ··· ar,σ(r)f(vσ(1), . . . , vσ(r)) σ over all possible choices σ, assigning to each integer from 1 to r an integer from 1 to n. Thus the sum is over all maps σ : {a, . . . , r} → {1, . . . , n}. We observe that if σ(i) = σ(j) for some i 6= j, then the corresponding term is 0 because f is alternating. Thus in our sum we may restrict to those σ which are 1-1. Our sum can be decomposed into a double sum, by grouping together all maps σ which send {1, . . . , r} into a given set S, and then taking the sum over all such sets S. Thus symbollically, we can write X X X = . σ S σ∈P (S) In each inner sum X a1,σ(1) ··· ar,σ(r)f(vσ(1), . . . , vσ(r)) σ∈P (S) we shuffle back the r-tuple (vσ(1), . . . , vσ(r)) to the standard position (vi1 , . . . , vir ), where i1, . . . , ir are the elements of S ordered so that i1 < ··· < ir. Then f changes by the sign εS(σ) and consequently each inner sum is equal to X εS(σ)a1,σ(1) ··· ar,σ(r)f(vS). σ∈P (S) Taking the sum over all possible sets S, we obtain the formula stated in the Theorem.  Theorem 2.2. Let V be a finite dimensional vector space over F , of dimension n. Let r be an integer. There exists a finite dimensional vector space T over F , and an r-multilinear alternating map ϕ : V (r) → T satisfying the following universal property. If U is a vector space over F and g : V (r) → U is an r-multilinear alternating map, then there exists a unique linear map

g∗ : T → U such that for all u1, . . . , ur ∈ V we have

g(u1, . . . , ur) = g∗(ϕ(u1, . . . , ur)). 7 Proof. If r > n the only alternating multilinear map g : V (r) → U to a vector space U is the zero map (for any w1, . . . , wr ∈ V , with r > n, one of the wi must be expressable as a linear combination of the others. Thus g(w1, . . . , wr) = 0). Thus T = {0} satisfies the universal property for r > n). Now assume that r ≤ n (actually, the following construction works “vacuously” for the case when r > n also). For each subset S of {1, . . . , n} consisting of precisely r elements, select a letter tS. Let T = V ({tS}) be the vector space over F consisting of formal linear combinations of the n {tS}. A basis of V ({tS}) is {tS}. The dimension of T is r . Let {v1, . . . , vn} be a basis of V . Let u1, . . . , ur be elements of V . Let A = (aij) be the matrix with coefficients in F such that u1 = a11v1 + ··· + a1nvn . . ur = ar1v1 + ··· + arnvn. Define ϕ : V (r) → T by X ϕ(u1, u2, . . . , ur) = DetS(A)tS. S ϕ is multilinear since each DetS is multilinear in the rows of A. ϕ is alternating since if ui = ui+1 for some i, then two adjacent rows of A are equal. Hence for each S, two adjacent rows of the submatrix of A corresponding to the set S are equal, and hence DetS(A) = 0.

Observe that tS = ϕ(vi1 , vi2 , ··· , vir ) if i1, . . . , ir are the elements of S, ordered so that i1 < ··· < ir. From the theorem on existence and uniqueness of linear maps having prescribed values on basis elements, we conclude that if g : V (r) → U is a multilinear alternating map, then there exists a unique linear map g∗ : T → U such that for each set S, we have

g∗(tS) = g(vS) = g(vi1 , . . . , vir ) if i1, . . . , ir are as above. By Theorem 2.1, it follows that

g(u1, . . . , ur) = g∗(ϕ(u1, . . . , ur)) for all elements u1, . . . , ur of V  (r) Theorem 2.3. Suppose that T1, with alternating multilinear map ϕ1 : V → T1, satisfies the universal property of Theorem 2.2, and T2, with alternating multilinear map ϕ2 : (r) V → T2, satisfies the universal property of Theorem 2.2. Then there exists a (unique) linear isomorphism ψ : T1 → T2 such that ψϕ1 = ϕ2. We denote the vector space satisfying the universal property of Theorem 2.2 by Vr V , and denote ϕ(u1, . . . , ur) by u1 ∧ · · · ∧ ur.

Proposition 2.4. Suppose that {w1, . . . , wn} is a basis of V . Then the set of elements Vr Vr {wi1 ∧ · · · ∧ wir } (1 ≤ i1 < ··· < ir ≤ n) is a basis of V . Hence V has dimension n r . Theorem 2.5. Let V be a finite dimensional vector space, and let r ≥ 1 be an integer. Then there is a unique linear map r r !∗ ^ ^ (V ∗) → V

8 given by

ϕ ∧ · · · ∧ ϕr 7→ Lϕ1∧···∧ϕr ∗ for ϕ1, . . . , ϕr ∈ V , where

Lϕ1∧···∧ϕr (v1 ∧ · · · ∧ vr) = Det(ϕi(vj)) Vr for v1 ∧ · · · ∧ vr ∈ V . This map is an isomorphism.

Proof. We first show that the desired linear map exists, and is unique. For fixed ϕ1, . . . , ϕr ∈ ∗ (r) V , the map V → F defined by (v1, . . . , vr) 7→ Det(ϕi(vj)) is multilinear and alternat- Vr ing, so there exists a unique linear map L(ϕ1,...,ϕr) : V → F defined by v1 ∧ · · · ∧ vr 7→ Det(ϕi(vj)). ∗ (r) Vr ∗ Now the map (V ) → ( V ) given by (ϕ1, . . . , ϕr) 7→ L(ϕ,...,ϕr) is multilinear and Vr ∗ Vr ∗ alternating, so there is a unique linear map (V ) → ( V ) given by ϕ1 ∧ · · · ∧ ϕr 7→

L(ϕ1,...,ϕr). This is the desired linear map. It remains to show that this map is an isomorphism. Since V and V ∗ have the same dimension, Vr(V ∗) and (Vr V )∗ have the same dimension. It thus suffices to show that the map is 1-1. ∗ ∗ ∗ Let {w1, . . . , wn} be a basis of V , with dual basis {w1, . . . , wn} of V .

{wβ1 ∧ · · · ∧ wβr | 1 ≤ β1 < β2 < ··· < βr ≤ n} is a basis of Vr V and ∗ ∗ {wα1 ∧ · · · ∧ wαr | 1 ≤ α1 < α2 < ··· < αr ≤ n} Vr ∗ is a basis of (V ). We compute that for 1 ≤ α1 < α2 < ··· < αr ≤ n and 1 ≤ β1 < β2 < ··· < βr ≤ n,  1 if α = β for 1 ≤ i ≤ r Det(w∗ (w )) = i i αi βj 0 otherwise. Suppose that r X ∗ ∗ ^ ∗ ϕ = cα1,...,αr wα1 ∧ · · · ∧ wαr ∈ (V ), 1≤α1<α2<···<αr≤n Vr ∗ with cα1,...,αr ∈ F , and the corresponding map L ∈ ( V ) is zero. Then L(wβ1 ∧ · · · ∧ wβr ) = 0 for all 1 ≤ β1 < β2 < ··· < βr ≤ n. Since L(wβ1 ∧ · · · ∧ wβr ) = cβ1,...,βr , we have that ϕ = 0. Thus the map is 1-1 and an isomorphism.  By the universal property, we have an isomorphism of (Vr(V ))∗ with multilinear alter- nating forms on V (r). The above theorem gives an identification of multilinear alternating forms on V (r) with Vr(V ∗). Theorem 2.6. Let V be a finite dimensional vector space over F . For each pair of integers r, s ≥ 1 there exists a unique bilinear map r s r+s ^ ^ ^ V × V → V such that if u1, . . . , ur and w1, . . . , ws are elements of V , then under this map we have

(u1 ∧ · · · ∧ ur) × (w1 × · · · ∧ ws) 7→ u1 ∧ · · · ∧ ur ∧ w1 ∧ · · · ∧ ws. 9 (s) Vr+s Proof. Given u1, . . . , ur ∈ V , the map V → V given by

(w1, . . . , ws) 7→ u1 ∧ · · · ∧ ur ∧ w1 ∧ · · · ∧ ws is s-multilinear and alternating. Hence there exists a unique linear map s r+s ^ ^ L(u1,...,ur) : V → V such that w1 ∧ · · · ∧ ws 7→ u1 ∧ · · · ∧ ur ∧ w1 ∧ · · · ∧ ws. The association

(u1, . . . , ur) 7→ L(u1,...,ur) is an r-multilinear map s r+s ^ ^ V (r) → L( V, V ) which is alternating, so there exists a unique linear map r s r+s ^ ^ ^ V → L( V, V ) denoted by ω 7→ Lω such that

u1 ∧ · · · ∧ ur 7→ L(u1,...,ur). The association (ω, η) 7→ Lω(η) is the desired linear map. 

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