Vector Space and Inner Product Space

Linear algebra is the study of linear maps on finite-dimensional vector spaces. Use of vectors to represent a countably infinite sequence is a small conceptual extension. Viewing waveforms as vectors is a larger conceptual extension. We have to view vectors as abstract objects rather than as n-tuples. Orthogonal expansions are best viewed in vector space terms. L2 waveforms can be viewed as vectors in the inner product space, such inner product space known as signal space.

Vector space

A vector space $\mathcal V$ is a set of elements, $\vec v \in \mathcal V$, called vectors, along with a set of rules for operating on both these vectors and a set of ancillary elements called scalars.
A vector space with real scalars is called a real vector space, and one with complex scalars is called a complex vector space.

Axioms of vector space

Addition

For each $\vec v \in \mathcal V$ and $\vec u \in \mathcal V$, there is a vector $\vec v + \vec u \in \mathcal V$ called the sum of $\vec v$ and $\vec u$ satisfying:

• Commutativity: $\vec v + \vec u = \vec u + \vec v$.
• Associativity: $\vec v + (\vec u + \vec w) = (\vec v + \vec u) + \vec w$, for each $\vec v, \vec u, \vec w \in \mathcal V$.
• Zero: there is a unique vector $0 \in \mathcal V$ such that $\vec v + 0 = \vec v$ for all $\vec v \in \mathcal V$.
• Negation: for each $\vec v \in \mathcal V$, there is a unique vector $−\vec v$ such that $\vec v + (− \vec v) = 0$.

Scalar multiplication

For each scalar $\alpha$ and each $\vec v \in \mathcal V$ there is a unique vector $\alpha \vec v \in \mathcal V$ called the scalar product of $\alpha$ and $\vec v$ satisfying:

• Scalar associativity: $\alpha (\beta \vec v) = (\alpha \beta)\vec v$ for all scalars $\alpha,\beta$, and all $\vec v \in \mathcal V$.
• Unit multiplication: for the unit scalar 1, $1\vec v = \vec v$ for all $\vec v \in \mathcal V$.

Distributive laws

• For all scalars $\alpha$ and all $\vec v, \vec u \in \mathcal V$, $\alpha (\vec v + \vec u) = \alpha \vec v + \alpha \vec u$.
• For all scalars $\alpha, \beta$ and all $\vec v \in \mathcal V$, $(\alpha + \beta)\vec v = \alpha \vec v + \beta \vec v$.

Finite-dimensional vector spaces

The set of finite-energy complex waveforms can be viewed as a complex vector space. For space $L^2$ of finite energy complex functions, we can define $\vec u + \vec v$ as function $\vec w$ where $w(t) = u(t) + v(t)$ for each t. Define $\alpha \vec v$ as vector $\vec u$ for which $u(t) = \alpha v(t)$.

• Span: A set of vectors $\vec {v_1}, \vec {v_2}, ..., \vec {v_n} \in \mathcal V$ spans $\mathcal V$ (and is called a spanning set of $\mathcal V$) if every vector $\vec v \in \mathcal V$ is a linear combination of $\vec {v_1}, \vec {v_2}, ..., \vec {v_n}$.
• Finite-dimensional: A vector space $\mathcal V$ is finite-dimensional if a finite set of vectors $\vec {v_1}, \vec {v_2}, ..., \vec {v_n}$ exist that span $\mathcal V$.
• Linearly dependent: A set of vectors $\vec {v_1}, \vec {v_2}, ..., \vec {v_n} \in \mathcal V$ is linearly dependent if $\sum_{j=1}^n \alpha_j\vec {v_j}$ for some set of scalars not all equal to 0. This implies that each vector $\vec {v_k}$ for which $\alpha_k \ne 0$ is a linear combination of the others.
• Basis: A set of vectors $\vec {v_1}, \vec {v_2}, ..., \vec {v_n} \in \mathcal V$ is defined to be a basis for $\mathcal V$ if the set both spans $\mathcal V$ and is linearly independent. The dimension of a finite-dimensional vector space is defined as the number of vectors in a basis.

Inner product spaces

Vector space definition lacks distance and angles. Inner product adds these features. The inner product of $\vec v$ and $\vec u$ is denoted $\langle\vec v, \vec u\rangle$.

Axioms of inner product space

• Hermitian symmetry: $\langle\vec v, \vec u\rangle = \langle\vec u, \vec v\rangle^*$
• Hermitian bilinearity: $\langle\alpha\vec v + \beta \vec u, \vec w\rangle = \alpha \langle\vec v, \vec w\rangle + \beta \langle\vec u, \vec w\rangle$; $\langle\vec v, \alpha \vec u + \beta \vec w\rangle = \alpha^* \langle\vec v, \vec u\rangle + \beta^*\langle\vec v, \vec w\rangle$
• Strict positivity: $\langle\vec v, \vec v\rangle \ge 0, \text{equality iff } \vec v = \vec 0 \tag{*}$.

For $\Bbb C^n$, we usually define $\langle\vec v, \vec u\rangle = \sum_i v_i u_i^*$.
If $\vec e_1, \vec e_2, ... , \vec e_n$are unit vectors in $\Bbb C^n$, then $\langle\vec v,\vec e_i\rangle = v_i, \langle\vec e_i,\vec v\rangle = v_i^*$.
$\|\vec v\|^2 = \langle\vec v, \vec v\rangle$ is squared norm of $\vec v$. $\|\vec v\|$ is length of $\vec v$.
$\vec v$ and $\vec u$ are are orthogonal if $\langle\vec v, \vec u\rangle = 0$. More generally $\vec v$ can be broken into a part $\vec v_{\bot\vec u}$ that is orthogonal to $\vec u$ and another part collinear $\vec v_{|\vec u}$ with $\vec v$.

One-dimensional projection theorem

Let $\vec v$ and $\vec u$ be arbitrary vectors with $\vec u \ne 0$ in a real or complex inner product space. Then there is a unique scalar $\alpha$ for which $\langle\vec v - \alpha\vec u, \vec u\rangle = 0$. That $\alpha$ is given by $\alpha = \langle\vec v, \vec u\rangle /\|\vec u\|^2$.

$${\vec v}_{|\vec u} = {\langle \vec v, \vec u \rangle \over \|\vec u\|^2} \vec u = \langle \vec v, {\vec u \over \|\vec u\|} \rangle {\vec u \over \|\vec u\|}$$ $$cos(\angle(\vec u, \vec v)) = \langle {\vec v \over \|\vec v\|}, {\vec u \over \|\vec u\|} \rangle {\vec u \over \|\vec u\|}$$

Pythagorean theorem: If $\vec v$ and $\vec u$ are orthogonal, then $\|\vec v + \vec u\|^2 = \|\vec v\|^2 + \|\vec u\|^2$.
Schwarz inequality: Let $\vec v$ and $\vec u$ be vectors in a real or complex inner product space, then $|\langle\vec v, \vec u\rangle| \le \|\vec v\| \|\vec u\|$.

The inner product space of $L^2$ functions

$L^2$ becomes an inner product space if we define the inner product of $L^2$ as

$$\langle \vec v, \vec u \rangle = \int_{-\infty}^{\infty} v(t) u^*(t) dt$$

Strict positivity axiom (*) does not hold for finite-energy waveforms, so we must define equality as $L^2$ equivalence.
The vectors in this space are equivalence classes. Alternatively, view a vector as a set of coefficients in an orthogonal expansion.

Vector subspaces

A subspace of a vector space $\mathcal V$ is a subset $\mathcal S$ of $\mathcal V$ that forms a vector space in its own right.
Equivalent: For all $\vec u, \vec v \in \mathcal V, \alpha \vec u + \beta \vec v \in \mathcal S$.
The notion of linear combination (which is at the heart of both the use and theory of vector spaces) depends on what the scalars are. So $\Bbb R^n$ is not a subspace of $\Bbb C^n$; real $L^2$ is not a subspace of complex $L^2$.
A subspace of an inner product space (using the same inner product) is an inner product space.

Reference:

1. Robert Gallager. (2006). 6.450 Digital Communication. MIT OpenCourseWare (http://ocw.mit.edu/)
2. Robert G.Gallager. (2009). Principles of Digital Communication (New York: Cambridge University Press).

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