1.10 Linear vector spaces

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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

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

• The vector-addition operation can be defined so that if $(x\land y\land z)\in S$ :
• $(x+y)\in S$ (the space is closed under addition)
• $x+y=y+x$ (Commutivity)
• $\left(x+y\right)+z=x+\left(y+z\right)$ (Associativity)
• The zero vector exists and is always an element of $S$ . The zero vector is defined by $x+0=x$ .
• For each $x\in S$ , a unique vector $-x$ is also an element of $S$ so that $x+-x=0$ , the zero vector.
• 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 forboth addition and multiplication. If $a$ , $b$ are scalars, the elements $x$ , $y$ of a linear vector space have the properties that:
• $ax$ (multiplication by scalar $a$ ) is defined and $ax\in S$ .
• $a\left(bx\right)=\left(ab\right)x$ .
• If "1" and "0" denotes the multiplicative and additive identity elements respectively of thefield of scalars; then $1x=x$ and $0x=0$
• $a(x+y)=ax+ay$ and $(a+b)x=ax+bx$ .

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

$\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ \\ {x}_{N}\end{array}\right)+\left(\begin{array}{c}{y}_{1}\\ {y}_{2}\\ \\ {y}_{N}\end{array}\right)=\left(\begin{array}{c}{x}_{1}+{y}_{1}\\ {x}_{2}+{y}_{2}\\ \\ {x}_{N}+{y}_{N}\end{array}\right)$
and scalar multiplication to be $a\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ \\ {x}_{N}\end{array}\right)=\left(\begin{array}{c}a{x}_{1}\\ a{x}_{2}\\ \\ a{x}_{N}\end{array}\right)$ . 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 $x(t)$ satisfies:

$\int_{{T}_{i}}^{{T}_{f}} \left|x(t)\right|^{2}\,d t$
One can verify that this collection constitutes a linear vector space. In fact, this space is so important that it hasa special name - ${L}^{2}({T}_{i}, {T}_{f})$ (read this as el-two ); the arguments denote the range of integration.

Let $S$ be a linear vector space. A subspace  of $S$ is a subset of $S$ which is closed. In other words, if $(x\land y)\in$ , then $(x\land y)\in S$ and all elements of  are elements of $S$ , but some elements of $S$ are not elements of  . Furthermore, the linear combination $(ax+by)\in$ for all scalars $a$ , $b$ . 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 thedistance between any two vectors can be obtained. The notions of length and distance are closely related to the concept ofan inner product.

An inner product of two real vectors $(x\land y)\in S$ , is denoted by $x\dot y$ and is a scalar assigned to the vectors $x$ and $y$ which satisfies the following properties:

• $x\dot y=y\dot x$
• $ax\dot y=a(x\dot y)$ , $a$ is a scalar
• $x+y\dot z=x\dot z+y\dot z$ , $z$ a vector.
• $x\dot x> 0$ unless $x=0$ . In this case, $x\dot x=0$ .

As an example, an inner product for the space consisting of column matrices can be defined as $x\dot y=x^Ty=\sum_{i=1}^{N} {x}_{i}{y}_{i}$ The reader should verify that this is indeed a valid inner product (i.e., it satisfies all of the properties givenabove). 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 theseproperties. For example, another valid inner product is $x\dot y=x^TKy$ where $K$ is an $NxN$ positive-definite matrix. Choices of the matrix $K$ which are not positive definite do not yield valid inner products ( property 4 is not satisfied). The matrix $K$ is termed the kernel of the inner product. When this matrix is something other than an identity matrix, the innerproduct is sometimes written as to denote explicitly the presence of the kernel in the inner product.

The norm of a vector $x\in S$ is denoted by $(x)$ and is defined by:

$(x)=(x\dot x)^{1/2}$

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

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

For the space $S$ consisting of column matrices, the norm of a vector is given by (consistent with the first choice of an inner product) $(x)=\sum_{i=1}^{N} {x}_{i}^{2}^{1/2}$ 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

$\left|x\dot y\right|\le (x)(y)$
This is one of the most important inequalities we shall encounter. To demonstrate this inequality, consider the normsquared of $x+ay$ . $(x+ay)^{2}=x+ay\dot x+ay=(x)^{2}+2a(x\dot y)+a^{2}(y)^{2}$ Let $a=-\left(\frac{x\dot y}{(y)^{2}}\right)$ . In this case: $(x+ay)^{2}=(x)^{2}-2\frac{\left|x\dot y\right|^{2}}{(y)^{2}}+\frac{\left|x\dot y\right|^{2}}{(y)^{4}}(y)^{2}=(x)^{2}-\frac{\left|x\dot y\right|^{2}}{(y)^{2}}$ 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 $x=-(ay)$ , or equivalently when $x=cy$ , where $c$ is any constant.

Two vectors are said to be orthogonal if the inner product of the vectors is zero: $x\dot y=0$ .

Consistent with these results is the concept of the "angle" between two vectors. The cosine of this angle is defined by: $\cos x,y=\frac{x\dot y}{(x)(y)}$ Because of the Schwarz inequality, $\left|\cos x,y\right|\le 1$ . The angle between the orthogonal vectors is $(\frac{\pi }{2})$ and the angle between vectors satisfying the Schwarz inequality with equality $(x, y)$ is zero (the vectors are parallel to each other).

The distance between two vectors is taken to be the norm of the difference of the vectors. $d(x, y)=(x-y)$

In our example of the normed space of column matrices, the distance between $x$ and $y$ would be $(x-y)=\sum_{i=1}^{N} ({x}_{i}-{y}_{i})^{2}^{1/2}$ which agrees with the Cartesian notion ofdistance. Because of the properties of the inner product, this distance measure (or metric ) has the following properties:

• $d(x, y)=d(y, x)$ (Distance does not depend on how it is measured.)
• $(d(x, y)=0)\implies (x=y)$ (Zero distance means equality)
• $d(x, z)\le d(x, y)+d(y, z)$ (Triangle inequality)
We use this distance measure to define what we mean by convergence . When we say the sequence of vectors $\{{x}_{n}\}$ converges to $x$ ( ${x}_{n}\to x$ ), we mean $\lim_{n\to }n\to$ x n x 0

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