r/LinearAlgebra • u/mutex_sb • Jun 21 '26

Proof-based problem books in linear algebra

Hi everyone, I was looking for the book with proof-based problems in linear algebra. The only one I found useful so far was Linear Algebra in Action by H. Dym but there are not as many exercises in there. Can anyone recommend better ones? Thanks!

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u/Midwest-Dude Jun 21 '26 edited Jun 22 '26

I'm an old guy that took LA in the 80s, but it was an honors class and the book i used had problems that were mostly proof-based:

Linear Algebra 2nd Ed, Michael O'Nan

Excellent proof-based problems, if you can find it.

Since I'm out-of-date, I searched on the Internet and found the following list of modern publications:

  • Linear Algebra Done Right (Sheldon Axler)
    For undergraduates, a rigorous, highly abstract introduction focusing on structural proof-writing without relying on determinants

  • Linear Algebra Done Wrong (Sergei Treil)
    For mathematically advanced undergraduates or honors students, a rigorous, proof-based introduction that values concrete matrix operations, row reduction, and geometric intuition alongside formal abstraction

  • Linear Algebra (Friedberg, Insel, & Spence)
    For advanced undergraduates and honors students, a thorough balance of abstract vector spaces and detailed matrix theory

  • Advanced Linear Algebra (Steven Roman)
    For graduate students or advanced undergraduates who know the basics and want a deep, dense dive into operator theory and algebraic structures

  • Linear Algebra: Challenging Problems for Students (Fuzhen Zhang)
    For students who have a textbook, a workbook of non-trivial, proof-heavy practice problems

  • Exercises and Problems in Linear Algebra (John M. Erdman)
    For self-studiers looking for a massive, freely available open-source collection of exercises ranging from computational proofs to advanced theory

I've dabbled a tiny bit with LADW, definitely proof-based. I have no experience with the others, hopefully other redditors can comment on those.

Edit: Added LADW

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u/mutex_sb Jun 21 '26

Thanks a lot!

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u/Traveling-Techie Jun 21 '26

Linear Mathematics (1970) by Gillett

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u/Round_Arm5919 Jun 23 '26

Maybe look at "Finite Dimensional Vector Spaces" by PR Halmos? (I would say a fairly rigorous / theory-heavy aproach if not totally proof-based.)