Contents |
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Preface |
Category Theory Primer |
Objects |
Morphisms |
Functors |
Dual Space |
Dual Basis |
Duality |
Connections with Category Theory |
Connections with Functional Analysis |
Preface
These are some notes on multilinear and geometric algebra, gathered from the following sources:
- Linear Algebra Done Right (5th Ed) by Sheldon Axler, particularly chapter 5.
- Mathematical Methods of Classical Mechanics (2nd Ed) by V.I. Arnold, particularly sections 32 and 33.
Category Theory Primer
These notes utilize the language of category theory.
Objects
A category $\cal C$ consists of a class $\text{ob}(\cal{C})$ of "objects" and a class $\text{hom}(\cal C)$ of "morphisms" that connect objects together and can be composed.
Note the term "class" and not "set" is used. This is because often times the classes of objects and morphisms are actually too large to even be described using sets. The term "class" is more appropriate since it is not constrained by set theory.
A prototypical example of a category is $\bf{Set}$, which is the category whose objects and the morphisms consists of maps between sets. The class of sets cannot itself be considered a set because "the set containing all sets" cannot be formed. This is due to "Russel's Paradox". Suppose $\text{ob}(\bf{Set})$ formed a set. And consider a subset consisting of all sets that do not contain itself as an element. Does this subset contain itself as an element? The answer leads to a paradoxical conclusion. In a similar way, the class $\text{hom}(\bf{Set})$ of all mappings cannot form a valid set.
The morphisms of a category are typically, but not always, structure-preserving maps. In linear algebra, for example, the category $\bf{Vect}$ consists of objects vector spaces who are vector spaces and whose morphisms are linear maps. And linear maps preserve the essential structure of a vector space: linear combinations.
There are plenty of other examples of categories from many branches of mathematics. But for now we concern ourselves with the following:
- $\bf{Vect}$: The afforementioned category of vector spaces.
- $\bf{Vect}(\mathbb K)$: The category of vector spaces over a specified field $\mathbb K$.
- $\bf{FinVect}(\mathbb K)$: The category of finite-dimensional vector spaces over a field $\mathbb K$.
Morphisms
A category is really about morphisms more than objects. Morphisms connect objects together and can be composed to create new morphisms. The following is the list of axioms of a category.
- Every object $X$ can be associated with its identity morphism $1_X$. In some formalisms of categories, objects are not primitivies but are rather synonymous with the identity morphisms.
- Each morphism $\phi$ can be associated with an object $\text{dom}(\phi)$ acting as its domain and an object $\text{cod}(\phi)$ acting as its codomain. The notation $\phi:X\to Y$ is meant to mean $X=\text{dom}(\phi)$ and $Y=\text{cod}(\phi)$.
- $\text{dom}(1_X)=\text{cod}(1_X)=X$.
- If $\phi:X\to Y$ and $\varphi:Y \to Z$ are morphisms then one can compose them into a morphism $\varphi\circ\phi:X\to Z$. And $1_Y\circ\phi=\phi\circ 1_X = \phi$.
Usually when morphisms are structure-preserving maps, then the identity morphisms is the identity map. This is the case for $\text{Set}$, $\text{Vect}$, and $\text{FinVect}$.
Morphisms may be categorized by special properties they may have:
- An endomorphism is a morphism $\phi$ for which $\text{dom}(\phi)=\text{cod}(\phi)$
- An isomorphism $\phi:X\to Y$ is a morphism for which there exists a morphism $\phi^{-1}:Y\to X$ for which $\phi\circ\phi^{-1}=1_Y$ and $\phi^{-1}\circ\phi=1_X$.
- An epimorphism is a morphism that is both an endomorphism and an isomorphism.
For $\text{Set}$, an isomorphism is any bijective function.
Functors
Functors are structure-preserving maps betwee two categories. There are two flavors of functors: covariant and contravariant.
A covariant functor $F$ associates objects and morphisms in a category $\cal C$ to a category $\cal D$ such that if $\phi:X\to Y$ is a morphism in $\text{hom}(\cal C)$, then $F(\phi):F(X)\to F(Y)$.
A contravariant functor $F$, however, reverses the domain and codomain: $F(\phi):F(Y)\to F(X)$.
Dual Space
In what follows, $V$ denotes an $n$-dimensional vector space over the field $\mathbb K$.
This vector space as a corresponding dual space, which is the set of all linear functions $V\to\mathbb K$. Elements of this space are known as linear functionals or covectors. The dual space is itself a vector space over $\mathbb K$ under the following rule of linear combinations:
Dual Basis
Let $e_1,\ldots,e_n$ form a basis of $V$. This means every $v\in V$ can be uniquely written as $v=v_1e_1+\ldots+v_ne_n$ for some coordinate tuple $(v_1,\ldots,v_n)\in\mathbb K^n$. Define the linear function $e^i$ ($1\leq i\leq n$) as the projection onto the $i$th axis: $e^i(v)=v_i$. It turns out that $e^1,\ldots,e^n$ forms a basis in $V^\star$ and that every functional $f\in V^\star$ may be written as $f=f(e_1)e^1 + \ldots + f(e_n)e^n$. This basis is the dual basis of $V^\star$.
Duality
Using the notation established in the previous subsection, the expression $f(v)$ may be written as
where $f_i=f(e_i)$ and $v_i = e^i(v)$. This formula is symmetric in the roles played by $f$ and $v$. They each act as the "coefficients" of the other. Indeed, $f(v)$ can equally be thought of as "appling $v$ to $f$" as much as "applying $f$ to $v$". This intuitive concept is known as duality.
To make this notional formal, consider the double dual space $V^{\star\star} = (V^\star)^\star$. This is the space of linear functionals on linear functionals. We show that elements in $V$ may be canonically identified with elements of $V^{\star\star}$. This is done via the evaluation map:
That is, $\text{eval}_V$ "applies" a vector $v$ to a functional $f$ to produce $f(v)$. By definition, $\text{eval}_V$ is a linear map. Moreover, it is an isomorphism. To establish injectivity and surjectivity, we just need to show that $\text{eval}_V(v)=0$ implies that $v=0$. The expression $\text{eval}_V(v)=0$ means that that $f(v)=0$ for all $f\in V^\star$. In particular, consider some dual basis $e^1,\ldots,e^n$ over $V^\star$. Then $v = e^1(v)e_1 + \ldots + e^n(v)e_n = 0$.
Connections with Category Theory
The vector spaces $V$, $V^\star$, and $V^{\star\star}$ are mutually isomorphic since they are all $n$-dimensional spaces over the same field $\mathbb K$. However, only $V$ and $V^{\star\star}$ are naturally isomorphic, with $\text{eval}_V$ being the "natural isomorphism".
Consider the "dual" functor $\star:\bf{FinVect}\to\bf{FinVec}$ mapping a finite-dimensional vector space $V$ to its duel $\star(V)=V^\star$ and each linear map $\phi:U\to V$ to its corresponding duel map $\star(\phi)=\phi^\star$:
This is a contravariant functor. The double-dual functor $\star\star:\bf{FinVect}\to\bf{FinVect}$ maps a finite-dimensional vector space $V$ to its double-duel $\star\star(V)=V^{\star\star}$ and a linear map $\phi:U\to V$ to its double-duel $\star\star(\phi)= (\phi^\star)^\star:U^{\star\star}\to V^{\star\star}$. This is a covariant functor.
Connections with Functional Analysis
In the preceding remarks on duality, $V$ was required to be finite-dimensional in order to show that $V$ and $V^\star$ were dually isomorphic. This does not generally hold when $V$ is not fininite dimensional.
However, there are some cases where $V$ is infinite-dimensional and its corresponding dual space $V^\star$ (defined as a subspace as the space of linear functionals on $V$) are dually isomorphic. This is the Riesz Representation Theorem, which is a foundational result of functional analysis.