Problems in Linear Algebra (1978), a book of solved problems | Hacker News Reader

Problems in Linear Algebra (1978), a book of solved problems

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160 pointshappy-go-lucky26 comments

I have the version of this book in Spanish (Editorial MIR), the translated title is 2000 Problems in Linear Algebra. This publishing house (MIR) doesn't exist anymore but now goes under the name of URSS [0] and publishes modern versions of these same books, at least in Spanish and Russian. It has several books on problem solving, which was probably a pathway to demonstrating true "elite" status in eastern europe at the time (hence the known romanian, hungarian and russian way of teaching). It also contains several books on theory too, be it mathematics, physics or engineering. The important thing with textbooks on theory is, unlike the custom of modern textbooks to have few problems and often disconnected or that require to make a few "leaps" in the sense that the exposition does not clearly pave a way to solve harder problems, the amount of exercises is large and they frequently build on top of one another, thus making learning much more enjoyable in my opinion.

Some books also worth looking at in mathematics:

(1) Anti-Demidovich series

(2) Combinatorial Analysis, Ribnikov

(3) Problem series by Suprun

(4) Probability Theory, Zolotarievskaya

(5) Discrete Mathematics Problems, Evnin

(6) Theory of Surfaces, Finikov

... and several other problem book series for university mathematics.

Engineering (mechanics):

(1) Anti-Mescherski series

(2) Theoretical Mechanics, V. M. Starzhinski

(3) Strength of Materials, Stiopin

(4) Problems in Strength of Materials, Volmir

[0] https://urss.ru/cgi-bin/db.pl?lang=sp&blang=en&page=Bookstor... (apparently the headquarters are in Peru now)

At a glance, probably Western equivalent of these are Schaum's Outline Series, which start by telling you something, giving you some worked examples on it, and then setting you lose on them. It's a great way to learn, quickly, particularly for a student.

I remember distinctly learning (as a teenager) Schaum's Outline Series of Vector Calculus teaching me quickly and in detail "grad div curl and all that" very effectively, so much so that I got basically 100% on all of my A-level maths and further maths papers because, well, if you can do a big book of problems like that, then exams really are just more of the same. At the time, I absolutely loved them -- big brown books, some of which are currently by my ankles.

I'm not sure it actually helped that much beyond early university though. Exams select for a very specific skill. What isn't taught is why something was discovered, or is useful, or how an idea came to be. Many more advanced ideas in mathematics are downright bizarre and it's the basic idea that you need to be able to come up with something similar to, not necessarily the detail.

I completely agree with you. A corollary of my comment is that while not necessarily providing theoretical justification, sometimes doing this whole problem solving thing helps seeing why a field of knowledge came to study some of its problems, fast. For example, I have a strong abstract foundation in Functional Analysis, Measure Theory and Optimization and my field is Numerical Analysis of PDE. I didn't understand 1. some mechanical properties of materials, 2. some properties of tensors and differential geometry that I needed. But I didn't have so much time to devote and work out the proofs for all the abstract apparatus on my own. So I took an engineering textbook on tensors, some notes on relativity (because they use tensors and Christoffel symbols) and in a few weeks I had almost understood everything I needed.

On the other hand, there is an extremely rigorous book with all the proofs for differential geometry, by Kennington. It has 2400 pages. For tensors I would guess that I need more Algebra, specifically, modules. And to review smoothness and differentiability classes.

Re: Kennington… I check in every year or so to see his progress.

For those interested: topology.org

By that book, do you mean the one authored by Spiegel?

I went over that book fully. From cover to cover. It was absolutely fantastic, and as good as a "problems book" can be.

I loved it and would recommend it to anyone who is looking to get their feet wet with problems in Vector Calculus.

Although it is neither a rigorous treatment (look into Arfken, Weber, Harris for that), nor it is greatly healpful pedagogically (look into Mary L. Boas for that), nor is it easy (look into Riley, Hobson, Bence for that).

The greatest intro material into Linear Algebra still remains the one by Gilbert Strang.

Yes, it was (is) the one by Spiegel! All of those are classic texts. Riley Hobson & Bence (or RHB) in particular was my first year of university text – it's very useful and well written, but I think it is not as "easy" as Boas. But for passing exams, Spiegel's problem book can't be beat :-).

Exactly.

I got perfect scores in both of the Mathematical Physics papers in college.

Many problems, with some variations did appear directly from Spiegel.

Best VC problems book ever.

For easier googling, the names of some of these authors in their slightly different English transliteration:

Rybnikov

Zolotarevskaya

Stepin

Note though that URSS is a ruthlessly commercial enterprise, and in the absence of any real competition, their printing is frequently bad, covers tasteless, translation and editing passable (which is better than can be said for more generalist Russian publishing houses’ attempts at technical literature) but only just, and prices high. (Before this February, the EUR prices before shipping were about 250% of the already-substantial RUB ones per the official conversion rates; nowadays I don’t know.) If you can at all help it, try the MCCME / IUM press and bookshop[1] first; unfortunately they do not deal in second-hand books (or rather they do, a bit, but people will pick those up in person in a day or two, before anybody even bothers entering them into the online catalogue).

Let me also add a recommendation for the calculus problems book by Günther (Gyunter) and Kuzmin (I don’t think it was ever translated, but how much translation do you really need in one?). It will not train you—every idea occurs once or maybe twice; it is not a book of exercises. (The joke goes that Demidovich is G&K with every problem repeated ten times.) But it is a book of problems, and it will teach you.

[1] https://biblio.mccme.ru/

I put together a list of the best linear algebra books, classified by type. Proskuryakov's practice book is one of them. Here's the full list:

https://begriffs.com/posts/2016-07-24-best-linear-algebra-bo...

Exercise: Let M be a square complex matrix satisfying M = M^T, and furthermore assume that M is not singular. Prove that there exists a matrix K such that K^2 = M and K = K^T.

Non-solution: You are essentially being asked to show that given an invertible complex-symmetric matrix, you can find a complex-symmetric square root. You might be tempted to use the Taylor series of sqrt(z) centred at some z_0 to obtain K, but this won't work if M's eigenvalues can't be fit inside a half-plane not include the complex number 0. An example of this is when M is

  ⎡1  0 ⎤
  ⎢     ⎥
  ⎣0  -1⎦

Another non-solution: You might attempt to extend the spectral theorem to complex-symmetric matrices, but the only such extension I know of is complicated, and results in a block-diagonal (but not necessarily diagonal) canonical form.

Solution: To actually extend the sqrt function from the complex numbers to matrices, use the Jordan canonical form of M. Then you only need to take the sqrt's of each Jordan block. Note that this sqrt won't exist if a Jordan block is non-zero and nilpotent, so we require M to be non-singular. This obtains K. Finally, observe that there exists a polynomial p such that p(M) = K (which can be obtained via Hermite interpolation), and therefore K^T = p(M)^T = p(M^T) = p(M) = K.

There's a geometric solution as well!

Since M is invertible and symmetric. It acts on R^N by independently scaling the coordinates of an orthogonal basis. Eg, M = O^T*A*O where O is orthogonal and A is a diagonal matrix. Let sqrt(A) be the diagonal matrix with entries equal to the square root of the corresponding entry in A. Then K = O^T*sqrt(A)*O is symmetric and K^2 = (O^T*sqrt(A)*O)*(O^T*sqrt(A)*O). Since O is orthogonal O^T*O = I, and K^2 = O^T*A*O = M.

That doesn't work. It's complex symmetric, but not real symmetric or Hermitian, so you can't use the spectral theorem.

This statement here:

  It acts on R^N by independently scaling the coordinates of an orthogonal basis

is precisely the spectral theorem, which you can't use here.

[edit]

Also, the corresponding claim for Hermitian (and real symmetric) matrices isn't true either. If an eigenvalue of a Hermitian matrix H is negative, then no square root of it can be Hermitian (or real symmetric).

Ah I missed the complex part. Clearly a complex matrix doesn't act on R^N by scaling the coordinates.

Your incorrect solution is the motivation for my "Another non-solution".

For real symmetric matrices, the n-dimensional coordinate scaling is still a nice visual picture of what's happening in the real case of the spectral theorem.

Paul Halmos wrote a book I like a lot titled "Linear Algebra Problem Book" that teaches you linear algebra through a sequence of problems - its different from the one posted here in that the problems are often preceded by 2-3 pages of discussion/motivation and almost all of them are conceptual and interesting.

Skimming through there are some interesting problems (the fibonacci sequence generated by determinants of nxn matrices with -1/1 on the lower subdiagonal/upper subdiagonal is pretty cool). But a lot of these problems just look like super tedious calculations.

> a lot of these problems just look like super tedious calculations

Analog Stack Overflow?

stackoverflow is a site for programming questions. Where does one see tedious mathematical calculations on there?

The math related stackoverflows

Mathoverflow is run by StackExchange.

https://mathoverflow.net/

That's the general style of the book, but further along the way it has problems that require proving statements by referring to the properties or definitions of the actual structures involved (vector spaces).

I've got 1938 problems but proving that the subspace L of the unitary space R_n has an orthogonal basis consisting of the eigenvectors of the transformation φ ain't one.

Boy, you've got a lot of problems.

99, to be precise.

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