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Linear Vector Spaces

Module by: Don Johnson

Summary: Introduces tools and formulas to use when dealing with Linear Vector Spaces. Topics covered include: linear vector spaces, inner product spaces, norm, Schwarz inequality, and distance between two vectors

One of the more powerful tools in statistical communication theory is the abstract concept of a linear vector space. The key result that concerns us is the representation theorem: a deterministic time function can be uniquely represented by a sequence of numbers. The stochastic version of this theorem states that a process can be represented by a sequence of uncorrelated random variables. These results will allow us to exploit the theory of hypothesis testing to derive the optimum detection strategy.

Basics

definition 1

A linear vector space S S is a collection of elements called vectors having the following properties:

  1. The vector-addition operation can be defined so that if x∧y∧z∈S x y z S :
    • x+y∈S x y S (the space is closed under addition)
    • x+y=y+x x y y x (Commutivity)
    • x + y +z=x+ y + z x + y z x y + z (Associativity)
    • The zero vector exists and is always an element of S S. The zero vector is defined by x+0=x x 0 x .
    • For each x∈S x S , a unique vector -x x is also an element of S S so that x+-x=0 x x 0 , the zero vector.
  2. Associated with the set of vectors is a set of scalars which constitute an algebraic field. A field is a set of elements which obey the well-known laws of associativity and commutivity for both addition and multiplication. If a a, b b are scalars, the elements x x, y y of a linear vector space have the properties that:
    • ax a x (multiplication by scalar a a) is defined and ax∈S a x S .
    • a b x = a b x a b x a b x .
    • If "1" and "0" denotes the multiplicative and additive identity elements respectively of the field of scalars; then 1x=x 1 x x and 0x=0 0 x 0
    • ax+y=ax+ay a x y a x a y and a+bx=ax+bx a b x a x b x .

There are many examples of linear vector spaces. A familiar example is the set of column vectors of length N N. In this case, we define the sum of two vectors to be:

x 1 x 2 ⋮ x N + y 1 y 2 ⋮ y N = x 1 + y 1 x 2 + y 2 ⋮ x N + y N x 1 x 2 ⋮ x N y 1 y 2 ⋮ y N x 1 y 1 x 2 y 2 ⋮ x N y N (1)
and scalar multiplication to be a x 1 x 2 … x N T=a x 1 a x 2 …a x N T a x 1 x 2 … x N a x 1 a x 2 … a x N . All of the properties listed above are satisfied.

A more interesting (and useful) example is the collection of square integrable functions. A square-integrable function xt x t satisfies:

∫ T i T f |xt|2dt<∞ t T i T f x t 2 (2)
One can verify that this collection constitutes a linear vector space. In fact, this space is so important that it has a special name - L 2 T i T f L 2 T i T f (read this as el-two); the arguments denote the range of integration.

definition 2

Let SS be a linear vector space. A subspace 𝒯𝒯 of SS is a subset of SS which is closed. In other words, if x∧y∈𝒯 x y 𝒯 , then x∧y∈S x y S and all elements of 𝒯𝒯 are elements of SS, but some elements of SS are not elements of 𝒯𝒯. Furthermore, the linear combination ax+by∈𝒯 a x b y 𝒯 for all scalars aa, bb. A subspace is sometimes referred to as a closed linear manifold.

Inner Product Spaces

A structure needs to be defined for linear vector spaces so that definitions for the length of a vector and for the distance between any two vectors can be obtained. The notions of length and distance are closely related to the concept of an inner product.

definition 3

An inner product of two real vectors x∧y∈S x y S , is denoted by <x,y> x y and is a scalar assigned to the vectors x x and y y which satisfies the following properties:

  1. <x,y>=<y,x> x y y x
  2. <ax,y>=a<x,y> a x y a x y , a a is a scalar
  3. <x+y,z>=<x,z>+<y,z> x y z x z y z , z z a vector.
  4. <x,x>>0 x x 0 unless x=0 x 0 . In this case, <x,x>=0 x x 0 .

As an example, an inner product for the space consisting of column matrices can be defined as <x,y>=xTy=∑i=1N x i y i x y x y i 1 N x i y i The reader should verify that this is indeed a valid inner product (i.e., it satisfies all of the properties given above). It should be noted that this definition of an inner product is not unique: there are other inner product definitions which also satisfy all of these properties. For example, another valid inner product is <x,y>=xTKy x y x K y where KK is an N x N N x N positive-definite matrix. Choices of the matrix KK which are not positive definite do not yield valid inner products (property 4 is not satisfied). The matrix KK is termed the kernel of the inner product. When this matrix is something other than an identity matrix, the inner product is sometimes written as x , y K x , y K to denote explicitly the presence of the kernel in the inner product.

definition 4

The norm of a vector x∈S x S is denoted by ∥x∥ x and is defined by:

∥x∥=<x,x>1/2 x x x 12 (3)

Because of the properties of an inner product, the norm of a vector is always greater than zero unless the vector is identically zero. The norm of a vector is related to the notion of the length of a vector. For example, if the vector x x is multiplied by the constant scalar a a, the norm of the vector is also multiplied by a a. ∥ax∥=<ax,ax>1/2=a∥x∥ a x a x a x 12 a x In other words, "longer" vectors ( a>1 a 1 ) have larger norms. A norm can also be defined when the inner product contains a kernel. In this case, the norm is written ∥x∥K K x for clarity.

definition 5

An inner product space is a linear vector space in which an inner product can be defined for all elements of the space and a norm is given by Equation 3. Note in particular that every element of an inner product space must satisfy the axioms of a valid inner product.

For the space S S consisting of column matrices, the norm of a vector is given by (consistent with the first choice of an inner product) ∥x∥=∑i=1N x i 21/2 x i 1 N x i 2 12 This choice of a norm corresponds to the Cartesian definition of the length of a vector.

One of the fundamental properties of inner product spaces is the Schwarz inequality

|<x,y>|≤∥x∥∥y∥ x y x y (4)
This is one of the most important inequalities we shall encounter. To demonstrate this inequality, consider the norm squared of x+ay x a y . ∥x+ay∥2=<x+ay,x+ay>=∥x∥2+2a<x,y>+a2∥y∥2 x a y 2 x a y x a y x 2 2 a x y a 2 y 2 Let a=-<x,y>∥y∥2 a x y y 2 . In this case: ∥x+ay∥2=∥x∥2-2|<x,y>|2∥y∥2+|<x,y>|2∥y∥4∥y∥2=∥x∥2-|<x,y>|2∥y∥2 x a y 2 x 2 2 x y 2 y 2 x y 2 y 4 y 2 x 2 x y 2 y 2 As the left hand side of this result is non-negative, the right-hand side is lower-bounded by zero. The Schwarz inequality is thus obtained. Note that the equality occurs only when x=-ay x a y , or equivalently when x=cy x c y , where c c is any constant.

definition 6

Two vectors are said to be orthogonal if the inner product of the vectors is zero: <x,y>=0 x y 0 .

Consistent with these results is the concept of the "angle" between two vectors. The cosine of this angle is defined by: cos x , y =<x,y>∥x∥∥y∥ x , y x y x y Because of the Schwarz inequality, |cos x , y |≤1 x , y 1 . The angle between the orthogonal vectors is ±π2 ± 2 and the angle between vectors satisfying the Schwarz inequality with equality x∝y ∝ x y is zero (the vectors are parallel to each other).

definition 7

The distance between two vectors is taken to be the norm of the difference of the vectors. dxy=∥x-y∥ d x y x y

In our example of the normed space of column matrices, the distance between xx and yy would be ∥x-y∥=∑i=1N x i - y i 21/2 x y i 1 N x i y i 2 12 which agrees with the Cartesian notion of distance. Because of the properties of the inner product, this distance measure (or metric) has the following properties:

  • dxy=dyx d x y d y x (Distance does not depend on how it is measured.)
  • dxy=0⇒x=y d x y 0 x y (Zero distance means equality)
  • dxz≤dxy+dyz d x z d x y d y z (Triangle inequality)
We use this distance measure to define what we mean by convergence. When we say the sequence of vectors x n x n converges to xx ( x n →x x n x ), we mean limn→∞∥ x n -x∥=0 n n x n x 0

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