Physics Derivation Graph: Math 402

Math 402

school: University of Missouri-Rolla (aka Missouri University of Science and Technology)
instructor: Barbara Hale
start date: 2007-08-30
end date: 2007-12-xx

Vector Spaces

Contents

Notation
Definitions of field, space, normed vector space, unitary vector space, orthogonality, linear independence, metric space, completeness basis vectors, dimension

Notation:

$\in$ = element of
$\exists$ = there exists
$∍$ = such that
$\forall$ = for every
iff = if and only if
$\vec{x}$ = vector
$α$ = complex number
$a$ = real number
$^{*}$ = complex conjugate
$\vec{r}$ = three dimensional position vector with components $(x, y, z)$

Definition of field:
$F \equiv {α, β, γ, . . .}$ where $α, β, γ, . . .$ are (in general) complex numbers and

$α + β$ and $α - β$ are defined and are elements of F. ( $α + β \in F$ and $α - β \in F$ .)
$α + (β + γ) = (α + β) + γ$ (Associative Property of addition) and
$α \cdot (β \cdot γ) = (α \cdot β) \cdot γ$ (Associative Property of multiplication) and
$α \cdot (β + γ) = (α \cdot β) + (α \cdot γ)$ (distributive property of multiplication over addition)
$α + β = β + α$ (commutative property of addition) and
$α \cdot β = β \cdot α$ (commutative property of multiplication)
The element $0$ exists where $a + 0 = a$ and
$a \cdot 0$ =0 and
$\forall α \in F$ there exists a $β$ such that $α + β = 0$ .
an identity, $E$ , exists such that $E \cdot α = α \forall α$ ; e.g., $E = 1$
at least one element of $F \neq 0$
$\forall α \in F, \exists β \in F ∍ α \cdot β = E$ ; e.g., $β \equiv α^{- 1}$
Read as "For every $α$ in field $F$ there exists $β$ in $F$ such that the product of $α \cdot β$ is element E (unity). In other words, $β$ is the inverse of $α$ . "

Definition of space:
$\vec{S} \equiv {\vec{x}, \vec{y}, \vec{z}, \vec{v}, . . .}$ where $\vec{x}, \vec{y}, \vec{z}, \vec{v}, . . .$ are mathematical objects ("vectors") over field $F$ and

$\vec{x} + \vec{y} \in \vec{S} \forall \vec{x} \in \vec{S}$ and $\vec{y} \in \vec{S}$ ; and
$α \vec{x} \in \vec{S} \forall α \in F$ and $\vec{x} \in \vec{S}$
$\vec{x} + \vec{y} = \vec{y} + \vec{x}$
$\vec{x} + (\vec{y} + \vec{z}) = (\vec{x} + \vec{y}) + \vec{z}$
$α (\vec{x} + \vec{y}) = α \vec{x} + α \vec{y}$
The "zero" or "null" vector $\vec{0}$ exists (and $\vec{0} \in \vec{S}$ ) $∍ \vec{x} + \vec{0} = \vec{x}$ and $α \vec{0} = \vec{0}$
also, $\forall \vec{x} \in \vec{S} \exists \vec{y} \in \vec{S} ∍ \vec{x} + \vec{y} = \vec{0}$ , e.g., $\vec{y}$ is the additive inverse of $\vec{x}$ .

Examples of vector spaces:

three dimensional Euclidean space. (The $\vec{r}$ coordinate space with $F$ being the real numbers.)
n-dimensional vector space over the field of complex numbers; $v e c x = (a_{1}, a_{2}, . . ., a_{n})$
set of all real, continuous functions $f (x)$ on $[0, 1]$ . Note that $f (x) = \vec{y}$ , a vector element of $\vec{S}$ .
set of all complex functions $Ψ (x)$ with domain $- \infty < x < \infty ∍ \int Ψ^{*} Ψ$ is finite
This is sometimes called $L^{2}$ , a Hilbert space of all square integrable functions
set of solutions to $\nabla^{2} f (\vec{r}) = 0$ , or $\nabla^{2} f (\vec{r}) = k^{2}$ for real $k$ .
set of functions $Ψ (\vec{r})$ where $| \vec{r} | \leq \infty$ and where the integral over all space $\int | Ψ (\vec{r}) |^{2} d^{3} x$ is finite.

n-dimensional vector space over the field of real numbers is described by $\begin{matrix} (1) & \vec{x} = x_{1} {\hat{e}}^{1} + x_{2} {\hat{e}}^{2} + . . . + x_{n} {\hat{e}}^{n} \end{matrix}$ where ${\hat{e}}^{1} = (1, 0, . . .0)$ and ${\hat{e}}^{2} = (0, 1, . . ., 0)$ . The ${\hat{e}}^{i}$ are called basis vectors.
A short-hand notation for $\vec{x}$ is $\begin{matrix} (2) & \vec{x} = (x_{1}, x_{2}, . . ., x_{n}) \end{matrix}$

A normed vector space is a vector space in which $\forall \vec{x} \in \vec{S}$ a quantity defined to be in the norm of $\vec{x}$ , denoted as $| | \vec{x} | |$ , exists. The norm must satisfy

$| | \vec{x} | | \geq 0$
$| | α \vec{x} | | = | α | \cdot | | \vec{x} | |$
$| | \vec{x} + \vec{y} | | \leq | | \vec{x} | | + | | \vec{y} | |$ . This is called the Minkowski inequality

| | \vec{x} | | = 0

iff

\vec{x} = \vec{0}

Example:
In an n-dimensional Euclidean space with $\vec{x} = (a_{1}, a_{2}, . . ., a_{n})$ then
$\begin{matrix} (3) & | | \vec{x} | | = \sqrt{| a_{1} |^{2} + | a_{2} |^{2} + . . . + | a_{n} |^{2}} \end{matrix}$

Unitary vector space, also known as a Hermitian vector space, also known as a complex inner product spaces.

A vector space is unitary iff it is possible to define a special operation called the inner product (or scalar product) $(\vec{x}, \vec{y}) \forall \vec{x}, \vec{y} \in \vec{S}$ . The inner product must satisfy

$(\vec{x}, \vec{y}) = (\vec{y}, \vec{x})^{*}$
$(\vec{x}, \vec{y} + \vec{z}) = (\vec{x}, \vec{y}) + (\vec{x}, \vec{z})$
$(\vec{x}, α \vec{y}) = α (\vec{x}, \vec{y})$
$(\vec{x}, \vec{x}) \geq 0$ and $(\vec{x}, \vec{x}) = 0$ iff $\vec{x} = \vec{0}$

From the above it can be shown that

\begin{matrix} (4) & (α \vec{x}, \vec{y}) = α^{*} (\vec{x}, \vec{y}) \end{matrix}

and

\begin{matrix} (5) & | (\vec{x}, \vec{y}) |^{2} \leq (\vec{x}, \vec{x}) \cdot (\vec{y}, \vec{y}) \end{matrix}

which is called the Cauchy-Shwartz Inequality.

Example
In an n-dimensional vector space, $\begin{matrix} (6) & (\vec{x}, \vec{y}) = x_{1}^{*} y^{1} + x_{2}^{*} y^{2} + . . . + x_{n}^{*} y^{n} \end{matrix}$

In matrix notation, $\begin{matrix} (7) & (\vec{x}, \vec{y}) = (x_{1}^{*}, x_{2}^{*}, . . ., x_{n}^{*}) \cdot [\begin{matrix} y^{1} \\ y^{2} \\ ⋮ \\ y^{n} \end{matrix}] = {\vec{x}}^{* T} \vec{y} \end{matrix}$ where $T$ is the transpose operation and $\vec{x}$ and $\vec{y}$ are matrices.

Definition of orthogonality:
Two vectors $\vec{x}$ and $\vec{y}$ are said to be orthogonal if $(\vec{x}, \vec{y}) = 0$ .

Definition: the vectors ${\vec{x}}_{1}, {\vec{x}}_{2}, . . ., {\vec{x}}_{n}$ are said to be linearly independent if
$\begin{matrix} (8) & α^{1} {\vec{x}}_{1} + α^{2} {\vec{x}}_{2} + . . . + α^{n} {\vec{x}}_{n} iff α^{i} = 0 \end{matrix}$ Any set of non-zero mutually orthogonal vectors are linearly independent.

A vector space is a metric space iff for every $\vec{x}$ and $\vec{y}$ in $\vec{S}$ it is possible to define a real number (written $d (\vec{x}, \vec{y})$ and called the metric) such that

$d (\vec{x}, \vec{y}) = 0$
$d (\vec{x}, \vec{y}) \leq d (\vec{x}, \vec{z}) + d (\vec{y}, \vec{z}) \forall \vec{z} \in \vec{S}$

Every normed vector space is a metric space with

d (\vec{x}, \vec{y}) = | | \vec{x} - \vec{y} | |

A normed vector space $\vec{S}$ is complete iff every Cauchy sequence of vectors ${{\vec{x}}_{n}}$

basis vectors,

dimension