Vector Spaces, Duals and Endomorphisms

Vector spaces, duals and endomorphisms

A real vector space V is a set equipped with an additive operation which is commutative and associative, has a zero element 0 and has an additive inverse −v for any v ∈ V (so V is an abelian group under addition). Further there is an operation of multiplication of the reals on the vectors (r, v) → rv ∈ V, for each real r and each vector v ∈ V, called scalar multiplication, which obeys, for any reals r and s and any vectors v ∈ V and w ∈ V, the relations: 0v = 0, 1v = v, (−1)v = −v, r(sv) = (rs)v,

(r + s)v = rs + sv, r(v + w) = rv + rw. The trivial vector space, said to be of zero dimensions, is a vector space consisting of only the zero vector. The basic family of non-trivial examples of a vector space are the spaces Rn, n ∈ N. Here Rn consists of all n-tuples x = (x1, x2, . . . , xn), with each xi real. The operations of Rn, valid for any reals xi, yi, i = 1, 2, . . . , n and any real s are:

x + y = (x1, x2, . . . , xn) + (y1, y2, . . . , yn) = (x1 + y1, x2 + y2, . . . , xn + yn),

x = (sx1, sx2, . . . , sxn). The zero element is 0 = (0, 0,..., 0) (often this element is just written 0). The additive inverse of x is −x = (−x1, −x2,..., −xn).

Given a pair of vector spaces V and W a map f : V → W is said to be linear if f(rx + sy) = rf(x) + sf(y), for any reals r and s and any x ∈ V and y ∈ V. In particular we have f(0) = 0 and f(−v) = −f(v), for any v ∈ V.

Denote the space of linear maps from V to W by Hom(V, W). If f and g are in Hom(V, W), their sum map is deﬁned by (f + g)(v) = f(v) + g(v), for any v ∈ V. Also we can multiply f ∈ Hom(V, W) by a real scalar r, giving the map rf, such that (rf)(v) = rf(v) for any v ∈ V. Then f + g and rf each lie in Hom(V, W) and these operations give Hom(V, W) the natural structure of a vector space. If f ∈ Hom(V, W) and g ∈ Hom(W, X), for vector spaces V, W and X, then the composition g ◦f : V → X is well-deﬁned and linear, so lies in Hom(V, X). Also the composition map (f, g) → f ◦ g is linear in each argument.

1 A linear map f ∈ Hom(V, W) is said to be an epimorphism if f is surjective, a monomorphism if f is injective (if and only if the equation f(v) = 0 has as its only solution the vector v = 0 and an isomorphism if f is bijective, in −1 −1 −1 which case f has an inverse f such that f ◦ f = idW and f ◦ f = idV are the identity maps (each of the latter is an isomorphism, each its own inverse). All trivial vector spaces are isomorphic. The space Hom(Rn, Rm) is isomorphic to the space Rmn. An element f ∈ Hom(Rn, Rm) is given by the n i Pn i j formula, for any x ∈ R , f(x) = y where y = j=1 fj x , i = 1, 2 . . . , m, for i i an m by n matrix fj . Then f is surjective if and only if the matrix fj has rank m and is injective, if and only if there are no solutions to the matrix i j i equation fj x = 0, except for the solution x = 0, i = 1, 2, . . . , n. Then f is an isomorphism if and only if m = n and the equation f(x) = 0 has as its only solution the vector x = 0, if and only if m = n and f has rank n.

A linear map from V to itself is called an endomorphism. Then the space Hom(V, V) of all endomorphisms of V is an algebra, with associative multiplication (distributive over addition) given by composition.

The space Hom(R, V) is naturally isomorphic to V itself: simply map f in Hom(R, V) to f(1) ∈ V.

The space Hom(V, R) is called the dual vector space of V and is written V∗. If f ∈ Hom(V, W) and α ∈ Hom(W, R), then f ∗(α) = α ◦ f is an element of Hom(V, R). As α ∈ W∗ varies, f ∗(α) depends linearly on α, so f ∗ gives a linear map from W∗ to V∗. The map f ∗ ∈ Hom(W∗, V∗) is called the adjoint of the map f. Then the map f → f ∗ is linear and (g ◦ f)∗ = f ∗ ◦ g∗, for any f ∈ Hom(V, W) and g ∈ Hom(W, X) and any vector spaces V, W and X. The adjoint of an identity map is itself, so the adjoint of an isomorphism is an isomorphism.

2 Bases and finite dimensionality A vector space is said to be finite dimensional if there is an isomorphism f : V → Rn, for some integer n. Any such isomorphism is called a basis for V. If v ∈ V and f is a basis, then f(v) ∈ Rn is called the co-ordinate vector of v in the basis f. The basis elements of Rn are the defined to be j the n-vectors, {ei, i = 1, 2, . . . , n}, such that the j-the entry of ei is δi , the Kronecker delta, so is 1 if j = i and zero otherwise.

Given a basis f : V → Rn of a vector space V, the corresponding basis −1 elements of V are the vectors {fi = f (ei), i = 1, 2, . . . , n}. Then for each 1 2 n Pn i v ∈ V, we have f(v) = v = (v , v , . . . , v ) if and only if v = i=1 v fi.

If f : V → Rn and g : V → Rm are bases for V, the map f ◦ g−1 : Rm → Rn is an isomorphism (with inverse g ◦ f −1), so m = n. So an n ∈ N, such that a basis f : V → Rn exists, is unique. It is called the dimension of V.

If f : V → Rn is a basis, then the adjoint f ∗ :(Rn)∗ → V∗ is an isomorphism, so (f ∗)−1 : V∗ → (Rn)∗ is an isomorphism. Now (Rn)∗ = Hom(Rn, R), so is isomorphic to Rn itself. This isomorphism maps t ∈ (Rn)∗ to the element n t = (t(e1), t(e2), . . . , t(en)) of R , where e1, e2, . . . , en are the standard basis elements for Rn. We call this isomorphism T . Then f : V → Rn is a basis, the map f T = T ◦ (f ∗)−1 : V∗ → Rn is a basis for V∗ called the dual basis to that of f. Then (f T )T = f. In particular V and V∗ have the same dimension.

If v ∈ V a vector space, then v gives an element v0 of (V∗)∗ = Hom(V∗, R) by the formula: 0 ∗ v (α) = α(v), for any α ∈ V Then the map V → (V∗)∗, v → v0 is an injection and is an isomorphism if V is ﬁnite dimensional.

Let V and W be vector spaces of dimensions n and m respectively. Let s : V → Rn and t : W → Rm be bases. Then if f ∈ Hom(V, W), deﬁne µ(f) = t ◦ f ◦ s−1. Then µ(f) ∈ Hom(Rn, Rm) so is represented by a matrix. If {si, i = 1, 2, . . . n} are the basis elements of V for the basis s and {tj, j = 1, 2, . . . m} are the basis elements of W for the basis t, then we have: Pm j j f(si) = j=1 fi tj, where fi is the matrix of µ(f). If v ∈ V has s-co-ordinate i Pn i j vector v, then f(v) has t-co-ordinate vector w, where w = j=1 fj v .

3 Tensor algebras

Let V be a real vector space. The tensor algebra of V, denoted T (V), is the associative algebra with identity over the reals, spanned by all the monomials of length k, v1v2 . . . vk, for all integers k, where each vi lies in V, subject to the relations of V: • If av + bw + cx = 0, with v, w and x in V and a, b and c real numbers, then aαvβ + bαwβ + cαxβ = 0, for any tensors α and β. If a tensor is a linear combination of monomials all of the same length k, the tensor is said to be of type k. The vector space of tensors of type k is denoted T k(V). We allow k = 0, in which case the tensor is just a real number. The tensors of type one are naturally identiﬁed with the vector space V itself.

If µ : W → V is a homomorphism of vector spaces, then there is a unique algebra homomorphism T (µ): T (W) → T (V), which reduces to µ when act- ing on W. Then µ maps each monomial w1w2 . . . wk to µ(w1)µ(w2) . . . µ(wk), for any w1, w2 . . . wk in W and any integer k. If also λ : X → W is also a vector space homomorphism, then we have: T (µ ◦ λ) = T (µ) ◦ T (λ).

n Using the standard basis elements {ei, i = 1, 2, . . . , n} of R , the tensor al- n gebra T (R ) has a natural basis given by the monomials ej1 ej2 . . . ejk where 1 ≤ jr ≤ n, for r = 1, 2, . . . , k and any k ∈ N, together with the number 1, the natural basis for tensors of type zero. Then every tensor of type k in T (Rn) can be written uniquely:

i1i2...ik T = T ei1 . . . eik . Here the Einstein summation convention is used: repeated indices are summed over.

If λ : V → Rn is a basis, so an isomorphism, then T (λ) gives an isomorphism of T (V) with T (Rn). In particular, every tensor T of type k in T (V) has a unique expression:

i1i2...ik −1 T = T fi1 . . . fik , where fi = λ (ei), i = 1, 2, . . . , n. The vector space T k(V) has dimension nk, for each nonnegative integer k.

Finally, it is sometimes necessary, for clarity, to use a notation for the tensor product operation: then TU is written T ⊗ U.

4 Covariant and contravariant tensors

Let V be a vector space of dimension n with dual space V∗. The full tensor algebra of V is the sub-algebra of the tensor algebra T (V ⊕ V∗) generated ∗ by monomials vi1 vi2 . . . vik such that each vi belongs either to V or to V . If a monomial is a product of p elements of V with q elements of V∗, then the tensor is said to be contravariant of type p and covariant of type q and of p type q . It is traditional to quotient out this tensor algebra by the relations:

∗ T vαU = T αvU, for any tensors T and U and and v ∈ V and any α ∈ V .

So the relative ordering of elements of V vis `avis elements of V∗ in any tensor monomial is immaterial.

If λ : V → W is a homomorphism of vector spaces, then the adjoint of λ, denoted λ∗ is the map λ∗ : W∗ → V∗ given by the formula:

∗ λ (β)(v) = β(λ(v)), for any v ∈ V. If λ is an isomorphism, then so is λ∗ and then the sum λ ⊕ (λ∗)−1 is an isomorphism, so induces an isomorphism T (λ ⊕ (λ∗)−1) of T (V ⊕ V∗) with T (W⊕W∗). In particular, if λ : V → Rn is a basis, so an isomorphism, every ∗ p tensor T of T (V ⊕ V ) of type q has a unique expression:

i i ...i T = T 1 2 p f . . . f f j1 . . . f jq . j1j2...jq i1 ip

Here the vectors fi ∈ V, i = 1, 2, . . . , n are determined by the formula, λ(fi) = j ∗ ei, i = 1, 2, . . . n and the dual vectors f ∈ V , j = 1, 2, . . . , n are determined by the relations f j(v) = λ(v)j, for any v ∈ V, or, equivalently, by the duality relations f j(f ) = δj. The quantities T i1i2...ip are called the components of i i j1j2...jq T . If U is another tensor, of type r, with components U k1k2...kr , then the s m1m2...ms tensor TU has components:

(TU)i1i2...ip+r = T i1i2...ip U ip+1ip+2...ip+r . j1j2...jq+s j1j2...jq jq+1jj+2...jq+s

5 Tensor contraction and multilinear maps

∗ p If τ = v1v2 . . . vpα1α2 . . . αq is a tensor monomial in T (V ⊕ V ) of type q , ∗ k with vi, i = 1, 2, . . . , p in V and αj, j = 1, 2, . . . q in V , then the l contraction of τ, denoted C k (τ), is the tensor: ( l )

C k (τ) = αl(vk)v1v2 . . . vk−1vk+1 . . . vpα1α2 . . . αl−1αl+1 . . . αq. ( l )

p Then C k extends to give a linear map from tensors of type to tensors ( l ) q p−1 k of type q−1 . This map is called the trace map over the index pair l . i1i2...ip i1i2...ik−1mik+1...ip If T has components Tj j ...j the trace C k (T ) has components: Tj j j mj ...j . 1 2 q ( l ) 1 2 l−1 l+1 q It follows that if S is a tensor of type p+q, with components Si1i2...... ip+q , then p+q j1j2...jp+q the tensor C 1 C 1 ...C 1 C 1 (S) (with p contractions) has compo- (q+1) (q+1) (q+1) (q+1) nents: Sm1m2...mpi1...iq . j1j2jqm1m2...mp Then the tensor: C 1 C 1 ...C 1 C 1 ...C 1 C 1 (S), with q contrac- (1) (1) (1) (q+1) (q+1) (q+1) tions of the kind C 1 and p contractions of the kind C 1 , has components: (1) (q+1) Sm1m2...mpn1...nq . In particular if T is of type p, with components T i1i2...ip n1n2nqm1m2...mp q j1j2...jq and U is a tensor of type q, with components U k1k2...kq , the complete con- p l1l2...lp traction of T and U is:

m1m2...mp n1n2...nq 1 1 1 1 1 1 T.U = C C ...C C ...C C (TU) = Tn n ...nq Um m ...mq . (1) (1) (1) (q+1) (q+1) (q+1) 1 2 1 2

p This complete contraction renders the vector spaces of tensors of types q q and p dual to each other. In particular, consider the special case that U is a j monomial: U = v1v2 . . . vqα1α2 . . . αp, for some vectors v ∈ V, j = 1, 2, . . . q ∗ and some co-vectors, αk ∈ V , k = 1, 2, . . . , p. Then put:

T (α1, α2 . . . αp, v1, v2 . . . , vq) = T.(v1v2 . . . vqα1α2 . . . αp).

Then this gives a well-deﬁned map from (V∗)p×Vq to the reals, which is linear in each variable, called a multilinear map. This gives a natural isomorphism p of the space of tensors of type q with the space of such multilinear maps.

6 1 Tensors of type 1 and endomorphisms 1 ∗ Let M be a tensor of type 1 , so an element of V ⊗ V . i i We have M = Mj f fj, in terms of basis elements {fi, i = 1, 2, . . . , n} of V j ∗ i and dual basis elements {f , j = 1, 2, . . . , n} of V . We call Mj the matrix of M. Then M has many interpretations: 1 • As a linear operator on tensors N of type 1 : N → M.N Here M.N is the complete contraction. i j In terms of components, M.N = Mj Ni = tr(MN), the trace of the matrix for M times the matrix for N.

• As a bilinear map from V × V∗ to the reals. This is the special case when N is a monomial vα, with v ∈ V and α ∈ V∗: M(v, α) = M.(vα).

i j In terms of components this is αiMj v . This is just the matrix product αMv, where α is regarded as a row matrix and v as a column matrix. i j Here v = v fi and α = αjf . 2 • As an endomorphism of V. This is the 1 contraction of the tensor product Mv. Equivalently it is the map: v → M(v, ).

i i j In terms of components this is v → Mj v . This is just the matrix i product Mv, where v = v fi is regarded as a column vector

∗ 1 • As an endomorphism of V . This is the 1 contraction of the tensor product αM. Equivalently it is the map: α → M(, α).

i In terms of components this is αj → αiMj . This is just the matrix j product αMv, where α = αjf is regarded as a row vector. i Finally the trace of M is the number M.I = Mi , where I is the identity 1 endomorphism of V. The trace itself is a linear operator on 1 tensors, so 1 as such is represented by a 1 tensor, called the Kronecker delta tensor, the identity endomorphism, whose trace I.I = n is just the dimension of the vector space V.

7 Tensors and non-commutative polynomials

A polynomial f(x) in n-variables xi, i = 1, 2, . . . , n is a linear combination of monomials xi1 xi1 . . . xik where 1 ≤ ij ≤ n, for each j = 1, 2, . . . , k and for any k, with real coeﬃcients. Usually one assumes that these variables commute, xixj = xjxi, for any i and j. Then polynomials form a commutative associative algebra under multiplication. However it is also possible to consider non-commutative polynomials where we make no assumption that the variable commute. This is natural, for example, when we consider polynomials in matrices. If now V is a vector space, we can associate to V a variable x which takes values in the dual vector space, V∗, such that to each vector v i we have the associated polynomial x(v). If v = v fi, with {fi, i = 1, 2, . . . , n} i a basis for V, we have x(v) = v xi, where xi = x(fi), i = 1, 2, . . . , n is a list of n-variables. Then by addition and associative multiplication, every tensor of T (V) may be thought of as a polynomial T (x) in the variable x, such that if it has type p and its components are T i1i2...ip , then the corresponding i1i2...ip polynomial is T (x) = T xi1 xi2 . . . xip . If T is thought of as a multilinear ∗ map, T (α1, α2, . . . , αp), with αi in V , the non-commutative polynomial of T is just: T (x) = T (x, x, . . . , x). Then the full tensor algebra of contravariant and covariant tensors based on the vector space V may be regarded as all polynomials T (x, y), where x takes values in V∗, y takes values in V and x commutes with y (but the x-variables do not commute with each other, nor do the y-variables commute with each other.

p If T (x, y) is of type q so has p x-variables and q y-variables, so can be written T (x, y) = T (x, x, . . . , x, y, y, . . . y) with p x-variables and q y-variables, denote by λk the operation of removing the k-th x-variable, so we have: λk(T )(x, y) = T (x, x, . . . , x, , x, . . . , x, y, y, . . . , y), which gives a vector-valued polynomial. Similarly, denote by λl the operation of removing the l-th y- variable, so we have: λl(T )(x, y) = T (x, x, . . . x, . . . , x, y, . . . , y, , y, . . . , y), l l l which gives a co-vector-valued polynomial. Then λk = λ λk = λkλ removes the k-th x-variable and the l-th y-variable. This gives a polynomial, denoted l ∗ 1 λk(T ), with values in V ⊗ V , the tensors of type 1 , given by the formula:

l λk(T )(x, x, . . . , x, y, y, . . . , y) = T (x, x, . . . x, , x, . . . , x, y, . . . , y, , y, . . . , y).

8 In terms of components we have:

i i ...i T (x, y) = T 1 2 p x x . . . x yj1 yj2 . . . yjq , j1j2...jq i1 i2 ip

i i ...i λ (T )(x, y) = T 1 2 p x x x x . . . x yj1 yj2 . . . yjq , k j1j2...jq i1 i2 ik−1 ik+1 ip i i ...i λl(T )(x, y) = T 1 2 p x x . . . x yj1 yj2 . . . y y . . . yjq , j1j2...jq i1 i2 ip jl−1 jl+1 i i ...i λl (T )(x, y) = T 1 2 p x x . . . x x . . . x yj1 yj2 . . . y y . . . yjq . k j1j2...jq i1 i2 ik−1 ik+1 ip jl−1 jl+1 Abstractly, since elements of V ⊗ V∗ have a trace, we can take the trace of l λk(T )(x, y). In terms of variables this is:

i i ...i mi ...i tr(λl (T ))(x, y) = T 1 2 k−1 k+1 p x x . . . x x . . . x yj1 yj2 . . . y y . . . yjq . k j1j2...jl−1mjl+1...jq i1 i2 ik−1 ik+1 ip jl−1 jl+1 We see that this operation is the non-commutative polynomial version of the tensor trace operation, C k . ( l )

9 From tensors to the symmetric and Grassmann algebras

Let T be a tensor algebra based on a real vector space V of dimension n. j Let V have basis {fi, i = 1, 2, . . . n}, with dual basis {f , j = 1, 2, . . . , n}. Then the elements of T may be regarded as polynomials in a non-commuting variable x which takes values in V∗. Contracting with the basis elements, the variable x gives rise to the variables {xi = x(fi), i = 1, 2, . . . , n}, so that we i k have x = xif . If T is a tensor of type 0 the corresponding polynomial is:

i1i2...ik T (x, x, . . . , x) = T xi1 xi2 . . . xik . Here T i1i2...ik = T (f i1 , f i2 , . . . f ik ) are the components of T in the basis {f i}. The multiplication of tensors is associative but not commutative.

We can force the multiplication to be commutative, if we like, by simply requiring that xixj = xjxi, for all i and j. This gives the symmetric tensor algebra ST , which is just an ordinary associative and commutative algebra of polynomials in n variables. Equivalently, we can take any tensor and symmetrize it: for s a permutation of the numbers 1, 2, . . . , p, and T is a tensor of type p, so has p co-vector arguments, denote the tensor with its arguments switched according to the permutation s by s(T ). There are p! such permutations, forming the symmetric group Sp. Then we deﬁne: X S(T ) = (p!)−1 s(T ),S(T ) S(U) = S(T ⊗ U).

s∈Sp This converts the tensor algebra into the symmetric tensor algebra.

Similarly, we can force the multiplication to be skew-commutative, if we like, by simply requiring that xixj = −xjxi, for all i and j. This gives the exterior or Grassmann algebra ΩT of T , which is associative. If T (x) and p q U(x) are of types 0 and 0 , we have the graded commutation rule: T (x)U(x) = (−1)pqU(x)T (x). Equivalently we can take any tensor T of type p and skew-symmetrize it: for s a permutation of the numbers 1, 2, . . . , p, denote by sgn(s) its sign. Then we deﬁne: X Ω(T ) = (p!)−1 sgn(s)s(T ), Ω(T ) ∧ Ω(U) = Ω(T ⊗ U).

s∈Sp This converts the tensor algebra into the Grassmann tensor algebra.