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


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


Definition of field:
\(F \equiv \{\alpha, \beta, \gamma, ...\}\) where \(\alpha, \beta, \gamma, ...\) are (in general) complex numbers and

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

Examples of vector spaces:

n-dimensional vector space over the field of real numbers is described by \begin{equation} \vec{x} = x_1 \hat{e}^1 + x_2 \hat{e}^2 + ... + x_n \hat{e}^n \end{equation} 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{equation} \vec{x} = (x_1, x_2, ..., x_n) \end{equation}

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

\(||\vec{x}||=0\) iff \(\vec{x}=\vec{0}\)

In an n-dimensional Euclidean space with \(\vec{x}=(a_1, a_2, ..., a_n)\) then
\begin{equation} ||\vec{x}||=\sqrt{|a_1|^2+|a_2|^2+...+|a_n|^2} \end{equation}

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

From the above it can be shown that \begin{equation} (\alpha\vec{x},\vec{y})=\alpha^*(\vec{x},\vec{y}) \end{equation} and \begin{equation} |(\vec{x},\vec{y})|^2 \leq (\vec{x},\vec{x})\cdot (\vec{y},\vec{y}) \end{equation} which is called the Cauchy-Shwartz Inequality.

In an n-dimensional vector space, \begin{equation} (\vec{x},\vec{y}) = x_1^*\ y^1 + x_2^*\ y^2 + ... + x_n^*\ y^n \end{equation}

In matrix notation, \begin{equation} (\vec{x},\vec{y}) = (x_1^*, x_2^*, ..., x_n^*)\cdot \begin{bmatrix} y^1 \\ y^2 \\ \vdots \\ y^n \end{bmatrix} = \vec{x}^{*T} \vec{y} \end{equation} 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{equation} \alpha^1\vec{x}_1 + \alpha^2\vec{x}_2 + ... + \alpha^n \vec{x}_n\ \texttt{iff}\ \alpha^i=0 \end{equation} 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

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,