Although the spaces V and V — have the same dimension, and are therefore

isomorphic, there is no natural map between them. The assignment eµ ’ e—µ

is unnatural because it depends on the choice of basis.

One way of driving home the distinction between V and V — is to consider

the space V of fruit orders at a grocers. Assume that the grocer stocks only

apples, oranges and pears. The elements of V are then vectors such as

x = 3kg apples + 4.5kg oranges + 2kg pears. (A.17)

Take V — to be the space of possible price lists, an example element being

f = (£3.00/kg) apples— + (£2.00/kg) oranges— + (£1.50/kg) pears— .

(A.18)

The evaluation of f on x

f (x) = 3 — £3.00 + 4.5 — £2.00 + 2 — £1.50 = £21.00, (A.19)

then returns the total cost of the order. You should have no di¬culty in

distinguishing between a price list and box of fruit!

We may consider the original vector space V to be the dual space of V —

since, given vectors in x ∈ V and f ∈ V — , we naturally de¬ne x(f ) to be

286 APPENDIX A. ELEMENTARY LINEAR ALGEBRA

f (x). Thus (V — )— = V . Instead of giving one space priority as being the set

of linear functions on the other, we can treat V and V — on an equal footing.

We then speak of the pairing of x ∈ V with f ∈ V — to get a number in the

¬eld. It is then common to use the notation (f, x) to mean either of f (x) or

x(f ). Warning: despite the similarity of the notation, do not fall into the

trap of thinking of the pairing (f, x) as an inner product (see next section) of

f with x. The two objects being paired live in di¬erent spaces. In an inner

product, the vectors being multiplied live in the same space.

A.3 Inner-Product Spaces

Some vector spaces V come equipped with an inner (or scalar) product. This

is an object that takes in two vectors in V and returns an element of the ¬eld.

A.3.1 Inner Products

If our ¬eld is the complex numbers, C, we will use the symbol x, y to denote

a conjugate-symmetric, sesquilinear, inner product of two elements of V . In

this string of jargon, conjugate symmetric means that

x, y = y, x — , (A.20)

where the “—” denotes complex conjugation, and sesquilinear3 means

x, »y + µz = » x, y + µ x, z , (A.21)

= »— x, z + µ— y, z .

»x + µy, z (A.22)

If our ¬eld is the real numbers, R, then the conjugation is redundant, and

the product will be symmetric,

x, y = y, x , (A.23)

and bilinear

x, »y + µz = » x, y) + µ x, z , (A.24)

»x + µy, z = » x, z + µ y, z . (A.25)

3

Sesqui is a Latin pre¬x meaning “one-and-a-half”.

A.3. INNER-PRODUCT SPACES 287

Whatever the ¬eld, we will always require that an inner product be non-

degenerate, meaning that x, y = 0 for all y implies that x = 0. A stronger

condition is that the inner product be positive de¬nite, which means that

x, x > 0, unless x = 0, when x, x = 0. Positive de¬niteness implies

non-degeneracy, but not vice-versa.

Given a basis eµ , we can form the pairwise products

eµ , eν = gµν . (A.26)

If the metric tensor gµν turns out to be gµν = δµν , we say that the basis

is orthonormal with respect to the inner product. We will not assume or-

thonormality without speci¬cally saying so. The non-degeneracy of the inner

product guarantees the existence of a matrix gµν which is the inverse of gµν ,

i.e. gµν g ν» = δµ .

»

If we take our ¬eld to be the real numbers, R, then the additional struc-

ture provided by a non-degenerate inner product allows us to identify V with

V — . For any f ∈ V — we can ¬nd a vector f ∈ V such that

f (x) = f , x . (A.27)

In components, we solve the equation

fµ = gµν f ν (A.28)

for f ν . We ¬nd f ν = g νµ fµ . Usually, we simply identify f with f , and hence

V with V — . We say that the covariant components fµ are related to the

contravariant components f µ by raising

f µ = g µν fν , (A.29)

or lowering

fµ = gµν f ν , (A.30)

the indices using the metric tensor. Obviously, this identi¬cation depends

crucially on the inner product; a di¬erent inner product would, in general,

identify an f ∈ V — with a completely di¬erent f ∈ V .

For vectors in ordinary Euclidean space, for which x, y ≡ x·y, the usual

“dot product”, there is another way to think of the operations of raising and

lowering indices. Given a vector x, we can consider the numbers

xµ = eµ , x . (A.31)

288 APPENDIX A. ELEMENTARY LINEAR ALGEBRA

These are called the covariant components of the vector x. If x = xµ eµ , we

have

xµ = eµ , x = eµ , xν eν = gµν xν , (A.32)

so the xµ are obtained from the xµ by the same lowering operation as before.

In an orthonormal basis, the covariant and contravariant components of a

Euclidean vector x are numerically coincident.

Orthogonal Complements

Another use of the inner product is to de¬ne the orthogonal complement4 of

a subspace U ‚ V . We de¬ne U ⊥ to be the set

U ⊥ = {x ∈ V : x, y = 0, ∀y ∈ U }. (A.33)

It is easy to see that this is a linear subspace. For ¬nite dimensional spaces

dim U ⊥ = dim V ’ dim U

and (U ⊥ )⊥ = U . For in¬nite dimensional spaces we only have (U ⊥ )⊥ ⊆ U .

A.3.2 Adjoint Operators

Given an inner product, we can use it to de¬ne the adjoint or hermitian

conjugate of an operator A : V ’ V . We ¬rst observe that for any linear

map f : V ’ C, there is a vector f such that f (x) = f , x . (To ¬nd it we

simply solve fν = (f µ )— gµν for f µ .) We next observe that x ’ y, Ax is

such a linear map, and so there is a z such that y, Ax = z, x . It should

be clear that z depends linearly on y, so we may de¬ne the adjoint linear

map, A† , by setting A† y = z. This gives us the identity

y, Ax = A† y, x

The adjoint of A depends on the inner product being used to de¬ne it. Dif-

ferent inner products give di¬erent A† ™s.

4

As an aside, we should warn you not to use the phrase orthogonal complement without

specifying an inner product. There is a more general concept of a complementary subspace

to U ‚ V , and this is perhaps what you have in mind. A complementary space is any

space W ∈ V such that we can decompose v = u + w with u ∈ U , w ∈ W , and with u, w

unique. This only requires that U © W = {0} (here “{0}” is the vector space consisting of

only one element: the zero vector. It is not the empty set) and dim U + dim W = dim V .

Such complementary spaces are not unique.

A.4. INHOMOGENEOUS LINEAR EQUATIONS 289

In the particular case that our chosen basis eµ is orthonormal, (eµ , eν ) =

δµν , with respect to the inner product, the hermitian conjugate A† of an

operator A is represented by the hermitian conjugate matrix A† which is