## 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

1. Notation
2. Definitions of field, space, normed vector space, unitary vector space, orthogonality, linear independence, metric space, completeness basis vectors, dimension
• $$\in$$ = element of
• $$\exists$$ = there exists
• $$\backepsilon$$ = such that
• $$\forall$$ = for every
• iff = if and only if
• $$\vec{x}$$ = vector
• $$\alpha$$ = complex number
• $$a$$ = real number
• $$^*$$ = complex conjugate
• $$\vec{r}$$ = three dimensional position vector with components $$(x,y,z)$$

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

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

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
$$\alpha\vec{x}\in\vec{S} \forall \alpha\in F$$ and $$\vec{x}\in\vec{S}$$
• $$\vec{x}+\vec{y}=\vec{y}+\vec{x}$$
• $$\vec{x}+(\vec{y}+\vec{z})=(\vec{x}+\vec{y})+\vec{z}$$
• $$\alpha(\vec{x}+\vec{y})=\alpha\vec{x}+\alpha\vec{y}$$
• The "zero" or "null" vector $$\vec{0}$$ exists (and $$\vec{0}\in\vec{S}$$) $$\backepsilon \vec{x}+\vec{0}=\vec{x}$$ and $$\alpha\vec{0}=\vec{0}$$
also, $$\forall \vec{x}\in\vec{S} \exists \vec{y}\in\vec{S}\backepsilon \vec{x}+\vec{y}=\vec{0}$$, 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; $$vec{x}=(a_1, a_2,...,a_n)$$
• 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 $$\Psi(x)$$ with domain $$-\infty \lt x \lt \infty \backepsilon \int \Psi^* \Psi$$ is finite
This is sometimes called $$L^2$$, a Hilbert space of all square integrable functions
• set of solutions to $$\nabla^2f(\vec{r})=0$$, or $$\nabla^2f(\vec{r})=k^2$$ for real $$k$$.
• set of functions $$\Psi(\vec{r})$$ where $$|\vec{r}|\leq\infty$$ and where the integral over all space $$\int|\Psi(\vec{r})|^2d^3x$$ is finite.

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

• $$||\vec{x}|| \geq 0$$
• $$||\alpha\vec{x}|| = |\alpha|\cdot||\vec{x}||$$
• $$||\vec{x}+\vec{y}||\leq||\vec{x}||+||\vec{y}||$$. 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{equation} ||\vec{x}||=\sqrt{|a_1|^2+|a_2|^2+...+|a_n|^2} \end{equation}

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

• $$(\vec{x},\vec{y})=(\vec{y},\vec{x})^*$$
• $$(\vec{x},\vec{y}+\vec{z})=(\vec{x},\vec{y}) + (\vec{x},\vec{z})$$
• $$(\vec{x},\alpha\vec{y})=\alpha(\vec{x},\vec{y})$$
• $$(\vec{x},\vec{x})\geq 0$$ and $$(\vec{x},\vec{x})=0$$ iff $$\vec{x}=\vec{0}$$
From the above it can be shown that \begin{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.

Example
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

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