Vector Spaces
Definition 31 (Vector space)
\(\def\vs#1{\mathsf{#1}}\)
A vector space over a field (Definition 30) \(F\) is a set, \(\vs{V}\) of elements called vectors on which addition,
\(\vec{u}+\vec{v}\) of vectors \(\vec{u}\) and \(\vec{v}\), is defined,
scalar multiplication, \(\lambda \vec{u}\) of a vector \(\vec{u}\) by a scalar \(\lambda\) from \(F\) is defined
and the following axioms hold:
Axiom 10 (Vector addition)
\[\vec{u}+\vec{v} \in \vs{V}\]
Axiom 11 (Assosciativity of addition)
\[(\vec{u} + \vec{v}) + \vec{w} = \vec{u}+(\vec{v}+\vec{w})\]
Axiom 12 (Commutativity of addition)
\[\vec{u}+\vec{v} = \vec{v}+\vec{u}\]
Axiom 13 (Existence of a zero vector)
\[\exists \vec{0} \in \vs{V} \mid \vec{u}+\vec{0} = \vec{u}=\vec{0}+\vec{u}\]
Axiom 14 (Existence of an additive inverse)
\[\forall \vec{u} \in \vs{V} \exists - \vec{u}\in \vs{V} \mid \vec{u}+(-\vec{u}) = \vec{0} = (-\vec{u})+\vec{u}\]
Axiom 15 (Existence of the scalar product)
\[\lambda \vec{u} \in \vs{V} \forall \lambda \in F\]
Axiom 16 (Distributivity of the scalar product)
\[\forall \vec{u}, \vec{v} \in \vs{V} , \forall \lambda, \mu \in F, \lambda(\vec{u}+\vec{v}) = \lambda \vec{u}+\lambda \vec{v}\]
The most common types of vectors encountered in physics, for example, are Euclidean vectors, which can be represented as tuples of (often real) scalars.
However, the notion of a vector space generalises this to any object which happens to satisfy the multiplication and addition requirements of the vector space.
Definition 32 (Vector Subspace)
A non-empty subset, \(w\), of a vector space \(\vs{V}\) over \(F\), such that
\[\label{eq:subspaceaddclose}
\vec{w}_1 + \vec{w}_2 \in w \qquad \forall \vec{w}_{1,2} \in w\]
\[\label{eq:scalarmultsubsclose}
\lambda \vec{w} \in w \qquad \forall \vec{w} \in w, \forall \lambda \in F\]
[Vector space sum] Let \(\vs{U}_1\) and \(\vs{U}_2\) be
subspaces of a vector space \(\vs{V}\), then the sum,
\(\vs{U}_1 + \vs{U}_2\) is defined,
\[ \begin{align}\begin{aligned} \label{eq:vectorspacesum}
\vs{U}_1 + \vs{U}_2 = \left\{ \vec{u}_1 + \vec{u}_2 \in \vs{V} \mid \vec{u}_1 \in \vs{U}_1 \wedge \vec{u}_2 \in \vs{U}_2 \right\}\\i.e. \ :math:`\vs{U}_1 + \vs{U}_2` is the set of vectors in\end{aligned}\end{align} \]
\(\vs{V}\), that, expressed as a vector in \(\vs{U}_1\) added to
a vector in \(\vs{U}_2\).
[Vector Direct Sum] A sum \(\vs{U}_1 + \vs{U}_2\) in which every
element can be expressed uniquely in the form
\(\vec{u}_1 + \vec{u}_2\), with \(\vec{u}_1 \in \vs{U}_1\), and
\(\vec{u}_2 \in \vs{U}_2\) is called a direct sum, and is denoted
\[\vs{U}_1 \oplus \vs{U}_2\]
The sum \(\vec{u}_1 + \vec{u}_2\) is the direct sum,
\(\vec{u}_1 \oplus \vec{u}_2\) iff
\(\vec{u}_1 \cap \vec{u}_2 = \{\vec{0}\}\)
[Linear Combination] Let \(\vec{u}_1, \vec{u}_2, \dots, \vec{u}_n\)
be vectors in the vector space \(\vs{V}\) over the field \(F\).
A linear combination of these vectors is a vector of the form
\[ \begin{align}\begin{aligned}\lambda_1 \vec{u}_1 + \lambda_2 \vec{u}_2 + \cdots + \lambda_n \vec{u}_n\\with :math:`\lambda_1, \lambda_2, \dots, \lambda_n \in F`.\end{aligned}\end{align} \]
[Span of a vector space] Let
\(\vec{u}_1, \vec{u}_2, \dots, \vec{u}_n\) be vectors in the vector
space \(\vs{V}\) over the field \(F\), then the subspace of
\(\vs{V}\) spanned by these vectors is denoted
\[ \begin{align}\begin{aligned}{\rm sp}(\vec{u}_1, \vec{u}_2, \dots, \vec{u}_n)\\and is defined by\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} \left\{ \lambda_1 \vec{u}_1 + \lambda_2 \vec{u}_2 + \cdots +
\lambda_n \vec{u}_n \mid \lambda_1, \lambda_2, \dots, \lambda_n \in
F \right\}\\So the supspace spanned by this sequence of vectors is the set of all\end{aligned}\end{align} \]
linear combinations which may be formed from the sequence.
[Finite Dimensional Vector Space] A finite dimensional vector space is
one which is spanned by a finite sequence of vectors.
[Linearly Independent Sequence] A sequence of vectors
:math:`vec{u}_1, vec{u}_2, dots, vec{u}_n in
vs{V}` is called a linearly independent sequence iff
\[\lambda_1 \vec{u}_1 + \lambda_2 \vec{u}_2 + dots + \lambda_n
\vec{u}_n = \vec{0}\]
is only possible when
\[ \begin{align}\begin{aligned}\lambda_1 + \lambda_2 + \cdots +\lambda_n = 0\\with :math:`\lambda_1,
\lambda_2, \dots , \lambda_n \in F`.\end{aligned}\end{align} \]
If \(\vs{W}\) is a subspace of \(\vs{V}\) such that it is
spanned by \(\vec{u}_1, \vec{u}_2, \dots, \vec{u}_n\), then there is
a subspace of this sequence which is linearly independent and still
spans \(\vs{W}\).
Bases
Definition 33 (Basis)
A basis is a linearly independent sequence of vectors which is a span of a vector space.
Suppose \(\vec{u}_1, \vec{u}_2, \dots, \vec{u}_n\) is a basis of a vector space \(\vs{V}\).
Then every element can be uniquely expressed as a linear combination of the sequence.
The unique scalar multiples of each are the components of the element \(\vec{x} \in \vs{V}\).
Suppose \(\vs{V}\) has a basis \(\vec{u}_1, \vec{u}_2 \dots \vec{u}_n\).
Then any sequence of vectors \(\vec{w}_1, \vec{w}_2, \dots \vec{w}_m \in \vs{V}\) with \(m > n\) is linearly dependent.
It is often common to encounter the bases of a vector space defining a coordinate system for the vector space.
Definition 34 (Dimension of a vector space)
Suppose \(\vs{V}\) is finite dimensional.
Then the dimension of \(\vs{V}\), denoted \(\dim(\vs{V})\), is the number of vectors in any basis of \(\vs{V}\).
Criterion 1 (Conditions for a basis)
A sequence of vectors in \(\vs{V}\) is a basis provided it possesses any two of the following conditions,
the sequence spans \(\vs{V}\)
the sequence is linearly independent
\(n = \dim(\vs{V})\)
Property 1 (Properties of a vector subspace)
Suppose \(\vs{V}\) is finite dimensional; let \(\vs{W}\) be a subspace of \(\vs{V}\), then,
\(\vs{W}\) is finite dimensional
\(\dim(\vs{W}) \leq \dim(\vs{V})\)
If \(\vs{W} \neq \vs{V}\) then \(\dim(\vs{W}) < \dim(\vs{V})\)
Any basis of \(\vs{W}\) can be extended to be a basis of \(\vs{V}\).