Name: Linear algebra through geometry
ISBN: 9780387975863

What Hacker News says

hintymadJan 2023

For what level and in which area? Books like _Methods of mathematical physics_ can be both too hard and irrelevant to your needs. For starters,

To become a better problem solver with high-school level maths:

  - Polya's How to Solve It.
  - Books of your choice about math contests.
  - Concrete Maths. I understand that this book is taught in college, but it requires very little advanced maths, and its techniques are hugely useful for high school students too.

To hone my intuitions. I learned it the hard way that college maths were different from high school math: in high school, my teachers painstakingly drilled intuitions into us with very targeted explanations and tons of well designed exercises. In college, we won't get such luxury. So, it's really up to us to understand mathematical concepts intuitively before diving into technical details. For that matter, the following books helped me a lot:

  - The visual series. Visual Complex Analysis and Visual Group Theory, for instance
  - Pinter's A book of Abstract Algebra
  - Strichartz's The Way of Analysis
  - Linear Algebra Through Geometry by Wermer. The book offers a comprehensive geometric interpretation to linear algebra concepts. It's especially helpful for me to understand quadratic forms.

To understand Analysis better. This area is vast, so I'll skip recommendations of excellent text books:

   - _Counterexamples in Analysis_. Those counterexamples in Analysis play a huge role in helping me truly appreciate the intricacies of Analysis. Similarly, books like _Counterexamples in Probability and Real Analysis_ are of great help too.
   - The Way of Analysis by Robert S. Strichartz. This books is AMAZING for laymen like me. You'd want someone to *explain* how concepts emerge, and how intuitions evolve.

To become good at maths by doing maths, so the following books used to help me a lot:

  - Problems and Proofs in Real Analysis
  - Putnam and Beyond. I still suck at maths, but those well designed problems in Putnam really taught me how to seek insights in higher maths.
  - Piotr's Problems in Mathematical Analysis. But really, any problem books that challenge you will do. I'd recommend you find problem sets from the website of university courses. They cover essential techniques, and will not be as overwhelming as the books.

thread

mathaftermathJul 2016

My favorite LA books are Linear Algebra by Friedberg/Insel[0] which is a combination of Axler style book with more computation oriented one (Terry Tao has a set of lectures based off this book). Another one I like is Modern Intro To LA by Henry Ricardo[1] which implicitly introduced me to Replacement theorem which is really overlooked in a ton of LA books. Again, this book's a rigorous mixture of both theory and computation done very well. If you've never seen higher level math before, there's Linear Algebra: Gateway to Mathematics by Robert Messer[2]. It has tons of commentary about elementary set theory and proof techniques along the way. Whenever someone mentions Axler's book, someone else brings up Treil's book. But there's a third one in the same league/group which is Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza[3]. Other favorites are Coding the Matrix by Philip Klein[4] for Python aficionados and Linear Algebra Through Geometry by Banchoff/Wermer[5] for those who like geometry.

If you are way beyond all this, you can still pick up new things from Advanced Linear Algebra by Steven Roman[6].

[0] https://www.amazon.com/Linear-Algebra-Stephen-H-Friedberg-eb...

[1] https://www.amazon.com/Modern-Introduction-Linear-Algebra/dp...

[2] https://www.amazon.com/Linear-Algebra-Mathematics-Robert-Mes...

[3] https://www.amazon.com/Linear-Algebra-Introduction-Mathemati...

[4] https://www.amazon.com/Coding-Matrix-Algebra-Applications-Co...

[5] https://www.amazon.com/Algebra-Through-Geometry-Undergraduat...

[6] https://www.amazon.com/Advanced-Linear-Algebra-Graduate-Math...

thread

rramadassSep 2025

I am not sure what you are trying to argue with this gish gallop of a comment filled with strawmen. The usefulness of Geometry in visualizing and building intuition in Mathematics is a given. Equally, its limitations when it comes to higher dimensions is also known. But for a beginning student (the context of this thread) it is the former which must be emphasized before he has developed the requisite mathematical maturity to deal with the latter. You do not want to scare away students from the important practical subject of Linear Algebra by focusing too much on its complexity in the initial stages of learning.

I highly recommend that you talk to/read the works of some mathematicians who are also involved with the research in the teaching of mathematics. There has been a lot of studies/research done on how to teach students build mathematical intuition using Geometry (and other graphical means) and in particular w.r.t. Linear Algebra. Some relevant references are given below for your edification; those by Harel are in particular, noteworthy.

The farin and hansford book mentioned earlier is one of the best introductory texts for beginners since it starts with building geometrical intuition for the study of the linear algebra. Pair it with some more rigorous text like for example; Linear Algebra Pure & Applied by Edgar Goodaire and the student can comfortably learn a pure mathematical approach with a clear understanding of what he is doing. Goodaire is a "pure mathematician" who specifically starts with 2D/3D graphs and then transitions to n-space with all necessary definitions/theorems/proofs included in a very accessible writing style. This is the ideal approach to the teaching and learning of Linear Algebra i.e. use geometry (and other graphical means) in conjunction with algebra, and where applicable, include caveats/limitations, but always with the aim and focus of teaching concepts and not mere computation.

References:

1) Intuition in Science and Mathematics: An Educational Approach by Efraim Fischbein - https://link.springer.com/book/10.1007/0-306-47237-6

2) The efficiency of visualization through geometry at mathematics education: a theoretical framework by Sefa Dundar et al. - pdf at https://www.sciencedirect.com/science/article/pii/S187704281...

3) Should we teach linear algebra through geometry? by Ghislaine Gueudet-Chartier - pdf at https://www.sciencedirect.com/science/article/pii/S002437950...

4) Geometric thinking in a n-space by Ghislaine Gueudet - pdf at https://shs.hal.science/hal-00529609/

5) Students' understanding of proofs: a historical analysis and implications for the teaching of geometry and linear algebra by Guershon Harel - pdf at https://www.sciencedirect.com/science/article/pii/S002437959...

6) Promoting Linear Algebraic Reasoning among Students: Affordances and Challenges by Guershon Harel - pdf at https://www.tandfonline.com/doi/full/10.1080/10511970.2024.2...

thread