Complex Vector Spaces

The functions tpi(x) are, in general, complex functions. As a consequence, ket space is a complex vector space, making it mathematically necessary to introduce a corresponding set of vectors which are the adjoints of the ket vectors. The adjoint (sometimes also called the complex conjugate transpose) of a complex vector is the generalization of the complex conjugate of a complex number. In Dirac notation these adjoint vectors are called bra vectors or bras and are denoted by or (/. Thus, the bra (0,j is the adjoint of the ket, ) and, conversely, the ket j, ) is the adjoint (0,j of the bra (0,j... 

This linear combination is clearly different from (3). The implication is that the two-dimensional vector space needed to describe the spin states of silver atoms must be a complex vector space an arbitrary vector in this space is written as a linear combination of the base vectors sf with, in general complex coefficients. This is the first example of the fundamental property of quantum-mechanical states to be represented only in an abstract complex vector space [55]. 

In terms of this notation the physical state of a system is represented in a dual complex vector space by kets, like a) and bras, (a. The state ket (or bra) carries complete information about the physical state. The sum of two kets is another ket. 

However there is another way to describe the moduli space which is called the ADHM description. It is quite relevant to us, so let us review on that. Let V, W he hermitian vector spaces whose dimensions are n, r. Define a complex vector space M by... 

Now we fix a Riemannian metric g which is invariant under the T-action. The symplectic form UJ together with the Riemannian metric g gives an almost complex structure I defined by uj v,x) = g Iv,w). With this almost complex structure, we regard the tangent space Tj-X as a complex vector space. Let X = be the decomposition into the... 

In this section we define and discuss complex vector spaces. We give many examples, especially of vector spaces of functions. Such vector spaces do not usually figure prominently in introductory courses on linear algebra, but the vector nature of functions is crucial in many areas of math and physics. Definition 2.1 Consider a set V, together with an addition operation... 

For example, the real line R is not a complex vector space under the usual multipUcation of real numbers by complex numbers. It is possible for the product of a complex number and a real number to be outside the set of real numbers for instance, (z)(3) = 3i R. So the real line R is not closed under complex scalar multiplication. 

The trivial complex vector space has one element, the zero vector 0. Addition is defined by 0 -I- 0 = 0 for any complex number c, define the scalar multiple of 0 by c to be 0. Then all the criteria of Definition 2.1 are trivially true. For example, to check distributivity, note that for any c e C we have... 

The simplest nontrivial example of a complex vector space is C itself. Adding two complex numbers yields a complex number multiplication of a vector by a scalar in this case is just complex multiplication, which yields a complex number (i.e., a vector in C). Mathematicians sometimes call this complex vector space the complex line. One may also consider C as a real vector space and call it the complex plane. See Figure 2.1. 

For every natural number n we can define a complex vector space... 

In another physics application, the Dirac equation for states of an electron in relativistic space-time requires wave functions taking values in the complex vector space C" = (ci, C2, c, C4) ci, C2, C3, C4 e C. These wave functions are called Dirac spinors. 

Now that we have defined complex vector spaces, we can introduce dimen-... 

Definition 2.2 Let V be a complex vector space. Let B be a finite subset of... 

Definition 2.3 A complex vector space is finite-dimensional if it has a finite basis. Any complex vector space that is not finite-dimensional is infinitedimensional. 

Suppose that V is a finite-dimensional complex vector space. By the definition this means that V has a finite basis. It turns out that all the different bases of V must be the same size. This is geometrically plausible for real Euclidean vector spaces, where one can visualize a basis of size one determiiung a line, a basis of size two determining a plane, and so on. The same is true for complex vector spaces. A key part of the proof, useful in its own right, is the following fact. 

Definition 2.4 Let V be a finite-dimensional complex vector space. Suppose that ui,..., u is a finite basis ofV. Then the dimension of V is n. 

Readers familiar with spin systems may recall that the study of spin yields a physical example of different bases for the same complex vector space. For instance, to study an electron, or any other particle of spin-1/2, one uses a basis of two kets. Which kets one chooses depends on the orientation of the Stern-Gerlach machine (real or imagined). One might use - -z> and — z) as a basis for one calculation and - -x> and — x) for another. No matter what... 

Let us calculate, for future reference, the dimension of the complex vector space of homogeneous polynomials (with complex coefficients) of degree n on various Euclidean spaces. Homogeneous polynomials of degree n on the real line R are particularly simple. This complex vector space is onedimensional for each n. In fact, every element has the form ex for some c e C. In other words, the one-element set x" is a finite basis for the homogeneous polynomials of degree n on the real line. 

In the end, dimension is important physically because we can associate a certain complex vector space to each orbital type, and the dimension of the complex vector space tells us how many different states can fit in each orbital of that type. Roughly speaking, this insight, along with the Pauli exclusion principle, determines the number of electrons that fit simultaneously into each shell. These numbers determine the structure of the periodic table. 

The notion of a linear transformation is crucial. A function from a (complex) vector space to a (complex) vector space is a (complex) linear transformation if it preserves addition and (complex) scalar multiplication. Here is a more explicit definition. 

Proposition 2.4 Suppose V is a finite-dimensional complex vector space and S is a subset ofV that spans V. Suppose W is a complex vector space. Suppose f S W is a function. Then there is a unique linear transformation T-. V -> VF such that for any s e S we have... 

We will reserve the plainer symbol P for homogeneous polynomials of degree n in only two variables, a star player in our drama.) The set P is a complex vector space of dimension 1 the set containing only the function f C, (x, y, z) 1 is a basis. Let T denote the restriction of the... 

Two important complex numbers associated to any particular complex linear operator T (on a finite-dimensional complex vector space) are the trace and the determinant. These have algebraic definitions in terms of the entries of the matrix of T in any basis however, the values calculated will be the same no matter which basis one chooses to calculate them in. We define the trace of a square matrix A to be the sum of its diagonal entries ... 

S on the complex vector space (with the usual basis). Unlike T, the linear... 

Proposition 2.11 Suppose V is a complex vector space of dimension zz e N. Suppose T . V V is a complex linear operator. Then T has at least one eigenvalue (and at least one corresponding eigenvector). 

This proof does not give a method for finding real eigenvalues of real linear operators, because the Fundamental Theorem of Algebra does not guarantee real roots for polynomials with real coefficients. Proposition 2.11 does not hold for inhnite-dimensional complex vector spaces eiffier. See Fxercise 2.28. 

Thus, for example, C" is equal (as a complex vector space) to the Cartesian sum of n copies of C ... 

Recall from Section 1.7 that the standard mathematical way to deal with irrelevant ambiguity is to define an equivalence relation and work with equivalence classes. In this case of tensors, the irrelevant ambiguity arises from the different ways of writing the same object as a linear combination of products. We will use this insight to define tensor products. Suppose V and W are complex vector spaces. Consider the complex vector space V W generated by the set... 

Because of the substitution rules in Definition 2.14, the complex vector space structure of V W descends to V W, so V IT is a vector space. 

Exercise 2.1 Consider the set of homogeneous polynomials in two variables with real coefficients. There is a natural addition of polynomials and a natural scalar multiplication of a polynomial by a complex number. Show that the set of homogeneous polynomials with these two operations is not a complex vector space. 

Exercise 2.3 Show that C (with the usual addition and multiplication) is itself a complex vector space of dimension 1. Then show thatC with the usual addition but with scalar multiplication by real numbers only is a real vector space of dim ension 2. 

Exercise 2.6 Let V be an arbitraty complex vector space of dimension n. Show that by restricting scalar multiplication to the reals one obtains a real vector space of dimension 2n. 

Exercise 2.12 Show that the set C2 of twice-differentiable complex-valued functions on R Zv a complex vector space. Find its dimension. Show that the Laplacian is a linear operator on C. ... 

Exercise 2.13 Suppose V is a complex vector space of finite dimension. Suppose W is a subspace ofV and dim W = dim V. Show that W = V. 

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