Multilinear Algebra

Contents
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:

Category Theory Primer

These notes utilize the language of category theory.

Objects

A category C \cal C consists of a class ob ( C ) \text{ob}(\cal{C}) of "objects" and a class hom ( C ) \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 S e t \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 ob ( S e t ) \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 hom ( S e t ) \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 V e c t \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:

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.

Usually when morphisms are structure-preserving maps, then the identity morphisms is the identity map. This is the case for Set \text{Set} , Vect \text{Vect} , and FinVect \text{FinVect} .

Morphisms may be categorized by special properties they may have:

For Set \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 F associates objects and morphisms in a category C \cal C to a category D \cal D such that if ϕ : X Y \phi:X\to Y is a morphism in hom ( C ) \text{hom}(\cal C) , then F ( ϕ ) : F ( X ) F ( Y ) F(\phi):F(X)\to F(Y) .

A contravariant functor F F , however, reverses the domain and codomain: F ( ϕ ) : F ( Y ) F ( X ) F(\phi):F(Y)\to F(X) .

Dual Space

In what follows, V V denotes an n n -dimensional vector space over the field K \mathbb K .

This vector space as a corresponding dual space, which is the set of all linear functions V K V\to\mathbb K . Elements of this space are known as linear functionals or covectors. The dual space is itself a vector space over K \mathbb K under the following rule of linear combinations:

( f + λ g ) ( v ) = f ( v ) + λ g ( v ) v V ; f , g V ; λ K \begin{gather*} (f+\lambda g)(v)=f(v)+\lambda g(v) \\ v\in V;f,g\in V^\star;\lambda\in\mathbb K \end{gather*}

Dual Basis

Let e 1 , , e n e_1,\ldots,e_n form a basis of V V . This means every v V v\in V can be uniquely written as v = v 1 e 1 + + v n e n v=v_1e_1+\ldots+v_ne_n for some coordinate tuple ( v 1 , , v n ) K n (v_1,\ldots,v_n)\in\mathbb K^n . Define the linear function e i e^i ( 1 i n 1\leq i\leq n ) as the projection onto the i i th axis: e i ( v ) = v i e^i(v)=v_i . It turns out that e 1 , , e n e^1,\ldots,e^n forms a basis in V V^\star and that every functional f V f\in V^\star may be written as f = f ( e 1 ) e 1 + + f ( e n ) e n f=f(e_1)e^1 + \ldots + f(e_n)e^n . This basis is the dual basis of V V^\star .

Duality

Using the notation established in the previous subsection, the expression f ( v ) f(v) may be written as

f ( v ) = f 1 v 1 + + f n v n , f(v)=f_1v_1 + \ldots + f_nv_n,

where f i = f ( e i ) f_i=f(e_i) and v i = e i ( v ) v_i = e^i(v) . This formula is symmetric in the roles played by f f and v v . They each act as the "coefficients" of the other. Indeed, f ( v ) f(v) can equally be thought of as "appling v v to f f " as much as "applying f f to v v ". This intuitive concept is known as duality.

To make this notional formal, consider the double dual space V = ( V ) V^{\star\star} = (V^\star)^\star . This is the space of linear functionals on linear functionals. We show that elements in V V may be canonically identified with elements of V V^{\star\star} . This is done via the evaluation map:

eval V : V V eval V ( v ) ( f ) = f ( v ) \begin{gather*} \text{eval}_V:V\to V^{\star\star} \\ \text{eval}_V(v)(f)= f(v) \end{gather*}

That is, eval V \text{eval}_V "applies" a vector v v to a functional f f to produce f ( v ) f(v) . By definition, eval V \text{eval}_V is a linear map. Moreover, it is an isomorphism. To establish injectivity and surjectivity, we just need to show that eval V ( v ) = 0 \text{eval}_V(v)=0 implies that v = 0 v=0 . The expression eval V ( v ) = 0 \text{eval}_V(v)=0 means that that f ( v ) = 0 f(v)=0 for all f V f\in V^\star . In particular, consider some dual basis e 1 , , e n e^1,\ldots,e^n over V V^\star . Then v = e 1 ( v ) e 1 + + e n ( v ) e n = 0 v = e^1(v)e_1 + \ldots + e^n(v)e_n = 0 .

Connections with Category Theory

The vector spaces V V , V V^\star , and V V^{\star\star} are mutually isomorphic since they are all n n -dimensional spaces over the same field K \mathbb K . However, only V V and V V^{\star\star} are naturally isomorphic, with eval V \text{eval}_V being the "natural isomorphism".

Consider the "dual" functor : F i n V e c t F i n V e c \star:\bf{FinVect}\to\bf{FinVec} mapping a finite-dimensional vector space V V to its duel ( V ) = V \star(V)=V^\star and each linear map ϕ : U V \phi:U\to V to its corresponding duel map ( ϕ ) = ϕ \star(\phi)=\phi^\star :

ϕ : V U ϕ ( f ) = f ϕ . \begin{gather*} \phi^\star:V^\star\to U^\star \\ \phi^\star(f)= f \circ \phi \end{gather*}.

This is a contravariant functor. The double-dual functor : F i n V e c t F i n V e c t \star\star:\bf{FinVect}\to\bf{FinVect} maps a finite-dimensional vector space V V to its double-duel ( V ) = V \star\star(V)=V^{\star\star} and a linear map ϕ : U V \phi:U\to V to its double-duel ( ϕ ) = ( ϕ ) : U V \star\star(\phi)= (\phi^\star)^\star:U^{\star\star}\to V^{\star\star} . This is a covariant functor.

( ϕ ) ( f ) ( g ) = f g ϕ . (\phi^\star)^\star(f)(g)=f \circ g \circ \phi.

Connections with Functional Analysis

In the preceding remarks on duality, V V was required to be finite-dimensional in order to show that V V and V V^\star were dually isomorphic. This does not generally hold when V V is not fininite dimensional.

However, there are some cases where V V is infinite-dimensional and its corresponding dual space V V^\star (defined as a subspace as the space of linear functionals on V V ) are dually isomorphic. This is the Riesz Representation Theorem, which is a foundational result of functional analysis.