# Text suggestion for linear algebra and geometry

I want to study more linear algebra over the summer, specifically relating it to geometry. I was originally going to read Shafarevich's Linear Algebra & Geometry, after a recommendation, but it has no exercises. Can anyone suggest a similar text? As for my related background, I learned linear algebra from Hubbard's Vector Calculus text, I've worked through most of Axler's LADR, and through chapter 5 or 6 of Artin.

• Kostrikin/Manin, "Linear Algebra and Geometry", books.google.de/books/about/… , sounds like a natural suggestion. Jun 11, 2013 at 17:41
• That text looks nice. Does anyone have a similar suggestion that is less expensive (or free)?
– user59083
Jun 11, 2013 at 18:40
• I found some answers to my question in answers to this post: math.stackexchange.com/questions/60918/… ... more suggestions are welcome.
– user59083
Jun 11, 2013 at 19:08
• You might find my answer to books about Linear algebra and geometric insight useful. Dec 16, 2014 at 14:52

In my opinion, having a basic knowlegde of algebra (Axler is very good, for sure), I would bet on learning different small topics from different books, because it is rather difficult to find everything in one book. If you want to have deeper insight in linear algebra with applications to geometry, I would suggest you to study the following (advanced) topics:

1. Tensor, symmetric and antisymmetric vector spaces (that is, given vector spaces $V$ and $W$, to construct the objects $V\otimes W$, $Sym^r(V)$, $\bigwedge^n V$) together with a review of basic algebra from a higher point of view. For this item, I have found an extremely good source, Multilinear Algebra and Applications. It will help you have a solid ground on linear algebra, being a quite nice book.

2. Basic Lie Theory: I strongly reccomend you Naive Lie Theory. It is a must for someone wishing to relate linear algebra and geometry. Exceptional good book.

3. (Optional) Clifford Algebras: this is a bit more difficult but nevertheless useful if you have time enough, at least to have a rough idea. Please read the article here and then try to read the first 20 pages of Spin Geometry.

I think this is a good planning to study in summer. The references are nice and easily readable, but also having a higher level.

• +1 for Iozzi's notes-I was completely unaware of them and they are quite good indeed! Dec 20, 2014 at 19:38

I spent nearly 3 years looking for an understanding as to how linear algebra related to geometry and how this approach was supposed to unify the subject, I have looked at every single one of the books mentioned here and none of them answered my question.

The best versions of what all these books are trying to do (in terms of relating all of linear algebra to geometry) are two rare books, one by Fekete "Real Linear Algebra", whose introduction absolutely must be read to get a sense on how to view linear algebra as a whole, and another by Dieudonné "Linear Algebra and Geometry", which is extremely geometric in spirit, uses very rigorous notation and whose introduction stresses the distinction between affine and metric properties of Euclidean geometry,

Comments such as Artin's famous comments on linear transformations, or the notion that a determinant should be interpreted as a homothety of volumes (best done in Dieudonné!) etc... make it seem like some deep unified view of linear algebra in terms of geometry exists, e.g. that something that explains all the theorems with pictures may exist. Similarly chapters such as "Unitary Geometry" in Weyl's 'Group Theory and Quantum Mechanics' would only lead one to push harder in finding some unified interpretation of linear algebra, so who wouldn't want to check every good reference?

I've since found there is only a partial answer to this question, and the answer is Gelfand, once you have general relativity and quantum mechanics to actually guide you into seeing this. You can basically view the first chapter (on vector spaces and inner product spaces) as developing a geometric formalism, modelled on putting a vector space into a curved space (manifold), applicable to general relativity (and Euclidean geometry by extension), and the second chapter (on operators and linear transformations) as developing an algebraic formalism, modelled on complex numbers and polynomials (which Axler also mentions, as I'm sure you've read) mainly applicable to quantum mechanics (remember QM is not going to demand pretty geometric interpretations! Hence the importance of discarding the necessity for geometric interpretations here, and it unifies the subject when one does this!).

In rough overview, in Ch. 1 you first begin discussing the affine geometry of parallelism through the concepts of linear independence, bases, changing bases & isomorphisms (refer to Dieudonne for a lovely reason to see why you can think of this as affine geometry, more generally it follows Klein's view of affine/Euclidean/projective geometry as a geometry invariant under parallel/orthogonal/central projections, but there is a more technical sense in which the word Affine is used in linear algebra (see below) so be careful), then we add perpendicularity via an inner product to get Euclidean geometry (or Hermitian geometry if you want complex numbers) and discuss the general principles, then we strictly focus on orthogonal geometry (i.e. w.r.t. an orthogonal basis) via Gram-Schmidt & least squares and stuff. After this linear, bilinear and quadratic forms enter the picture, which is just another (general) way to talk about non-orthogonal (curvilinear) geometry, it just allows you to do Euclidean, Hermitian, Minkowskian & Symplectic geometry in one fell swoop using the same ideas, and then the final sections discuss Lagrange's & Jacobi's methods for reducing a quadratic form to a sum of squares, which is just another way of saying Einstein's equivalence principle, i.e. that locally at any point of spacetime we can work as if we are working in an orthogonal geometry, but globally this will not be true, i.e. we can diagonalize our metric (a quadratic form) locally using these methods, but globally it will not hold! (The very last section is on Hermitian geometry, i.e. doing all this stuff over the complex numbers). Thus the picture is all motivated by imagining putting a vector (tangent) space to a curved manifold and invoking the equivalence principle locally. You can ignore all this and pretend we're studying the algebra for it's own sake as the other books do, but you don't get a unified motivation/explanation for what you're doing that way...

There is a similar way to naturally motivate all the Jordan normal forms, eigenvalues, adjoint, self-adjoint, normal, unitary etc... concepts of linear algebra & I'll write a big explanation up if you like this description so far.

Based on all this, I can now mention a more advanced version of Gelfand & Shafarevich, namely Kostrikin, who has an exercises book does the affine stuff the way it's usually done & really does link operators with geometry (!), but I recommend this view as a secondary outgrowth/application of the algebraic interpretation to actually carrying out the Euclidean, Hermitian, Minkowskian & Symplectic geometry I mentioned above, because it doesn't unify the subject the way thinking about complex numbers/polynomials does. Basically you can view sections from chapter 2 of Kostrikin as like a chapter 3 of Gelfand mixing your ideas together, or perhaps just extended versions of his section on bilinear forms.

Thus, I recommend a mix of Dieudonné, Fekete, Gelfand & Kostrikin + exercise book. Hope this helps!

For more on the geometry, take a look at Berger's wonderful two-volume text Geometry. It's a very sophisticated treatment of classical geometries (not differential geometry), full of linear algebra. You might also look at Pedoe's beautiful book, Geometry, A Comprehensive Course, which uses multilinear algebra as well as linear algebra. There's also the beginnings of Lie groups, so you could look at Curtis's Matrix Groups.

• +1 for a very good list of uncommon sources-especially Berger,which sadly is ridiculously expensive for the average student and is a must have for those interested in a modern take on classical geometry via linear algebra. Dec 20, 2014 at 19:40

As darij grinberg comments above, there's Linear Algebra and Geometry by Suetin, Kostrikin, and Manin; it's fairly difficult, but it should be accessible, given the time between now and when you originally asked this question. I didn't read all of it, but quite liked what I did. It has plenty of good exercises.

On a different level entirely (and not helpful for you, given that you've read Hubbard and Axler - I'm mostly putting this here in case someone else runs into this question) is Ted Shifrin's Linear Algebra: A Geometric Approach is a nice geometric approach to linear algebra. It's less abstract than the sources you give (and covers far less ground than Shafarevich's book does), but offers a lot of geometric intuition. He also prides himself on his exercises, in contrast to your experience with Shaferevich.

The best general advanced linear algebra book I know is Module Theory An Approach to Linear Algebra by T.S.Blyth. It's beautifully written, very careful and modern.It has probably the most detailed treatment of multilinear algebra you'll find outside of a graduate algebra text. It may be a bit too difficult for your level, though. Check it out and judge for yourself.

I'm happy to report there are several excellent advanced books on the subject now available from Dover. A very good introduction to the geometry of linear algebra is Linear Algebra and Geometry: A Second Course by Irving Kaplansky. This is a strongly rigorous and abstract treatment by one of the masters of algebra of the last century. it focuses largely on the geometry of inner product and projective spaces,which are very naturally expressed in terms of linear transformations. I think you'll find this book very helpful indeed. Also from Dover and written by one of the true masters, is Linear Algebra and Projective Geometry by Rienhold Baer. It's considerably more difficult and specialized then Kaplansky, but I doubt you'll find a deeper treatment of the connection between these 2 important subjects. Lastly, one of the most comprehensive treatments of the relationship between classical geometry and abstract algebra can be found in Groups and Symmetry by Paul Yale. This is a surprisingly sophisticated treatment of not only groups of transformations,but the relations between rings and algebras and the classical transformations as well. A wonderful treatment you simply have to have and there's no good reason not to.

• -1 for no reason at all. 'Tis the season for my trolls to give me agita........... Dec 22, 2014 at 20:52

A bit different look, to build different intuitions:

Paul R. Halmos, A Hilbert Space Problem Book.

It cannot be the main textbook for you, but seems to be a good source of exercises on the border of linear algebra and geometry, from near elementary to very advanced ones.

• Uh-I think that's not only too sophisticated for the reader's level, but more then that,it's only tangentially related to the subject matter requested. Dec 20, 2014 at 19:34

N.Jacobson. Lectures in Abstract Algebra, vol 2. Linear Algebra.

M.M. Postnikov. Lectures in Geometry: Semester II. Linear Algebra and Differential Geometry.

Linear Algebra and Geometry Shafarevich, Igor R., Remizov, Alexey Translated by Kramer, D.P., Nekludova, L. 2013, http://www.springer.com/mathematics/algebra/book/978-3-642-30993-9

• The third is what I was seeking an alternative to. Why do you recommend the first two?
– user59083
Dec 18, 2014 at 5:38
• @5space People in MSE are not reading the body of the question anymore. They read only what you put in the title. I've noticed this a lot in my questions. Dec 20, 2014 at 6:09

I remember when I was in your situation trying to find the right source for good studying and intuitive thinking.

I recommend this MIT course with full video lectures, notes, problem sets, practice tests, and challenge problems (the best!). I personally like this course as a whole because it develops you intuition over you reasoning, which is what a mathematician needs. In addition, I have also learned from this course about a six months or so ago and with full knowledge and understanding of the subject of mathematics. I have done all the practice problems and even the challenging problems and that is what guided me through the course fast and efficiently. The course took me about a month or so to complete if you work really hard every day and constantly do problems so that the theorems and definitions just "stick."

Another recommendation is the edX course by The University of Texas at Austin (UTAustinX). It is actually a live online course that lets you interact with the community of other students in the program and the professors directing the course (which to me is quite amazing). the course features a real virtual classroom setting with homework and tests and even a certificate of completion in the end (if you make it to the end which I know you will :) ) I would recommend that you begin this course after you have gone through half of the MIT course. You have time; course starts on January $28, 2015.$

Another great source is the MIT OpenCourseWare. There are over $20$ different versions of teaching linear algebra by MIT professors. However, no matter which one you choose, you will have a great experience with the concepts.

In reference to learning geometry, I believe these course hold a firm foundation of high geometry. If you take a look at them and experience the course yourself, you will understand. However, if you would like a supplementary source specific to geometry, I think that your current answers have pretty good suggestions for that.

My final remarks: The internet is full of many great sources that you can choose from. If you ever have trouble or find my resources not helpful (I hope not!), then click here for what I did to search linear algebra resources. Again, no matter what you choose from my "arsenal," I bet that you will have a great intuitive experience with these sources. Also, it is always good to have more than one source when you are studying because you can compare and contrast the ideas presented by each other resource.

I understand that you are higher level. So I would recommend clicking here. You will be able to learn analytic geometry along with linear algebra in the first result. In addition, I will say that there are over a million results. You will have many graduate and undergraduate courses and lecture notes to work with to further you studies.

If you have anymore questions, just feel free to email me. Good luck with your studies!

P.s., I am sorry that I did not put it in a numbered list for you. I thought that it would be better in a paragraph format for both of our benefit, as I wanted to write a lot to make sure that you understand where each source can from and their format.

• The material you suggest is on the elementary level. OP asks for more advanced texts and with connection to geometry.
– Leo
Dec 18, 2014 at 9:55
• Please let me know why I got down voted. Dec 19, 2014 at 0:19

For linear algebra I recommend Linear Algebra and its Applications by Peter Lax.

A free online book A First Course in Linear Algebra by Robert A. Beezer