The missing linear algebra for pure mathematics course

\[\def\ev{\mathrm{ev}} \def\span#1{\operatorname{span}\{#1\}} \def\Hom{\operatorname{Hom}}\]

Recently I came across this ams bulletin which on the second page loosely refers to “all the classical constructions of linear algebra.” Things like the dual space, direct sums, tensor products, alternating powers, etc. Then, given a differential operator $D$ on a space $V$, it extends the operator to all of these constructions and defines the differential Galois group as all the field automorphisms which fix $D$-preserved subspaces of these “classical constructions.”

Anyway, none of those differential galois theory details matter here. What struck me is that almost every “classical construction of linear algebra” mentioned there is absent from almost every linear algebra class. In fact, the language of linear algebra used throughout the article, and in every pure mathematics class I took after basic algebra, is absent from linear algebra courses. The treatment of linear algebra in higher math is one of those things you just pick up as you go — but it does not have to be.

Linear algebra is perhaps the most important field in (pure and applied) mathematics. Much of the game of mathematics is reducing your problem to something in linear algebra and then solving it there. Entire subfields of mathematics could be characterized as generalizations of linear algebra concepts: Representation theory generalizes eigenspace decomposition, homological algebra generalizes the rank-nullity theorem, functional analysis is characterized as infinite-dimensional linear algebra. Considering all this, motivating a new treatise on the subject isn’t hard at all.

The two canonical treatments of linear algebra are Gilbert Strang’s 18.06 on MIT OCW, and Sheldon Axler’s Linear Algebra Done Right textbook. Strang’s treatment looks like most courses in linear algebra. It works entirely in coordinates and with operations on row/column vectors and matrices. Axler’s treatment introduces abstract vector spaces and linear maps. Both work over fields $F = \mathbb{R}, \mathbb{C}$.

One might think that this is already a sufficient treatment. Strang’s course represents the course given to all STEM students and has a more applied flavor. Axler’s is the more abstract treatment that only the pure math students get. Though as a pure math student who learned from both Strang’s and Axler’s approaches, I was continually surprised by how different linear algebra was in the more advanced courses I took.

I would have to pick up tidbits like working over finite fields, $\Hom (V,W)\simeq V^\ast\otimes W$, and trace being the composition \(\Hom (V,V)\simeq V^\ast\otimes V \xrightarrow{\ev} F\). Constructions like the tensor product, symmetric product, and anti-symmetric product were new to me. Ditto for the tensor up-restriction of scalars adjunction. Many classical constructions were entirely foreign to me.

Truthfully, the problem is that there is no introductory linear algebra text written in today’s language of pure mathematics. Axler bills his book as a second treatment in linear algebra that “will focus on abstract vector spaces and linear maps.” The course/text I am picturing would focus on operations on vector spaces in a category of vector spaces. That is, it would take the role of being the missing course on category theory too.

In this respect, one could think of this as the third and final treatment of linear algebra: First Strang, then Axler, and finally this one. This would reflect how I learned linear algebra. Though I would want the text to be self-contained, as Strang and Axler are, and I would picture it being a good second treatment of linear algebra. It would help to have seen something like Strang or Axler beforehand.

The Style of the Course (or book)

The typical linear algebra course looks like this:

• Coordinate focused: row/column vectors and matrices
• All vector spaces are finite-dimensional
• Base-field $F = \mathbb{R}$ or $\mathbb{C}$

This treatment would differ on all accounts:

• A primarily coordinate-free treatment
• Finite and Infinite Dimensional Vector Spaces
• Beyond $F = \mathbb{R}, \mathbb{C}$
• Additional Topics

A primarily coordinate-free treatment

It may come as a surprise to some, but all of the following can be defined without reference to a row/column vector or matrix:

• Vectors, Vector Spaces
• Bases, Rank
• Transpose, Adjoint
• Upper Triangular Matrices/Diagonal matrices
• Symmetric/Anti-symmetric matrices
• Inner Products
• Trace
• Determinant
• Tensors
• Eigenvectors, eigenvalues, eigenspaces
• Jordan Canonical Form

Lets do a few examples now to illustrate: trace, transpose, upper triangular matrices, and Jordan Canonical form.

Trace: There is a canonical map $V^\ast\otimes V\rightarrow \Hom_k(V,V)$ and $V^\ast \otimes V\xrightarrow{\ev} k$. The former sends the pair $\phi\otimes v$ to the rank $1$ linear map $w\mapsto \phi(w) v$. The latter sends the pair $\phi\otimes v$ to $\phi(v)$. Composing them one recovers the trace.¹ \(\text{tr}: \Hom_k(V,V)\rightarrow V^\ast \otimes V\rightarrow k\)

Transpose: The transpose is the functorial part of the dual-space functor. Given $V\xrightarrow{f} W$, one gets a map $W^\ast \xrightarrow{f^\ast} V^\ast$. If $W$ is finite-dimensional, we also have $W\simeq W^{\ast\ast}$ and using that isomorphism we know that $\Hom(V,W)\simeq V^\ast\otimes W\simeq (W^\ast)^\ast\otimes V^\ast\simeq \Hom(W^\ast, V^\ast)$ and this composition of isomorphisms yields the transpose.

Upper Triangular Matrices: Given a linear map $f\in \Hom_k(V,V)$, how could we express that this map is upper-triangularizable without talking about the matrix of $f$? Well, what does it mean to be upper triangularizable in a conceptual way? If we have an upper-triangular matrix of an endomorphism, we know that the left-most column is an entry with a bunch of zeroes. This means the vector $(1,0,0,\ldots,0)^t$ is sent to $(a,0,0,\ldots,0)^t$ - it is an eigenvector. Moreover, if we let $V_1 = \text{Span}_k((1,0,0,\ldots,0)^t)$, this subspace of $V$ is preserved by the linear map $f$. Similarly, we know that $\text{Span}_k ((1,0,\ldots)^t, (0,1,0,\ldots)^t)$ is preserved. Lets call this subspace $V_2$. Continuing, one finds a map is upper-triangularizable if there exists a sequence of vector subspaces

\[V_0\subset V_1\subset\cdots \subset V_n = V\]

such that $f$ preserves each $V_i$. Such a decomposition of a vector space is called a (complete) flag.²

Jordan Canonical Form: Given a linear endomorphism $f\in \Hom_k(V,V)$, for $V$ $n$-dimensional, one can decompose $V$ into $\bigoplus V_{\lambda_i}$ where $V_{\lambda_i} = \text{ker}(f-\lambda_i)^n$. Thus each $f-\lambda_i$ restricts to, and is nilpotent on, each $V_{\lambda_i}$. If one takes the subring $k[f]\subset \Hom_k(V,V)$, the max-spectrum will correspond to the different $V_{\lambda_i}$s and the degree at each point will correspond to the degree of nilpotency.

Finite and Infinite Dimensional Vector Spaces

While most of the examples will be finite-dimensional - infinite dimensional examples will play a role. $k^\infty$ in the form of $k[x]$, $k^\omega$ for its dual, examples from algebra, and some examples from functional analysis.

Beyond $F = \mathbb{R}, \mathbb{C}$

We will do linear algebra over

• $\mathbb{R}$
• $\mathbb{C}$
• $\mathbb{Q}$
• $\mathbb{Q}(i)$ - a few (likely all quadratic) number fields
• $\mathbb{F}_p$, $\mathbb{F}_q$ - finite fields
• $\mathbb{Z}$, $k[x]$, PIDs

That is right, with the last few examples we will even touch on modules to some extent - though only with the nicest and most linear algebraic examples. This is not meant to be a full algebra course but it certainly could aid one. Doing linear algebra with euclidean domains and principal ideal domains gets one’s feet wet.

Additional Topics

There will be some algebra covered here and there. Groups and group actions will be defined, as will rings and ideals. However the treatment will not be deep. Similarly, I would like to include a few chapters giving primers on the many fields which are derived from central concepts in linear algebra. Following is a potential chapter list:

The Table of Contents:

• Chapter 0: Life in the Category of Sets
• Chapter 1: Vector Spaces, Rings, Fields, and Modules
• Chapter 2: Spectral Theory of finite-dimensional Vector Spaces
• Chapter 3: Multilinear algebra³
• Chapter 4: Linear Algebra over Principal Ideal Domains
• Chapter 5: Inner Product Spaces
• Primer: Representation Theory of Finite Groups
• Primer: Functional Analysis
• Primer: Fourier Analysis
• Primer: Homological Algebra
• Special Topics: Algorithms in Linear Algebra

In chapter 0, I’d introduce universal properties and give many examples of them in set. I’d frame it as a new framing of many set-theory concepts I hope the students had seen before. Though if one were teaching a course it would probably help to jump in/out of chapter 0. Chapters 1-5 would be a core linear algebra course with much of the language categories used throughout. Chapter 1 would also introduce those “classical constructions” that I had hoped to have seen more before. Then there would be five entirely optional chapters for further reading in linear algebra that can play a part in courses given how much time is left.

Concluding thoughts

I hope none of this is taken as a reproach of Strang or Axler. I very much so enjoyed learning from both fo these resources - and this course/text would certainly be missing a lot of the content from Strang and Axler. Though this is a reproach of the literature in general. Though the approach of picking up various things haphazardly as one learns is an essential skill—much of life is built on this skill—I think the absense of a solid reference on this material is not something that should continue.

The closest resources I have found out there that achieve these goals would be Evan Chen’s Math55a notes from when Dennis Gaitsgory taught the course in 2014 and Aluffi’s celebrated textbook Algebra: Chapter 0. Both of these follow the language and style of what I am looking for, but the former is a little too little and the latter is a little too much. It’d be great to have something in the middle.

For my part, I don’t know if I will try to teach a course like this, make a youtube lecture series in this style, or write the textbook I am looking for. I could see myself doing all three. Though maybe drawing some attention to this issue will help some.

Footnotes