/sci/ - Science & Math

[Anonymous No.11353689]

What's a good book that covers the basics of geometry founded on linear algebra? By this I mean a book that develops geometry starting from the notion of a (real or complex) vector space, without relying (implicitly or explicitly) on synthetic axiomatic geometry (based on Hilbert's axioms or something equivalent) or even worse on heuristic geometric intuition.

I tried reading "Algebra and Geometry" by Alan F. Beardon but I'm a bit disappointed by it. Just by looking at the table of contents I can tell the book covers a lot of interesting topics, but I feel it is a little lacking in rigor. I can gloss over the informal proof of the fundamental theorem of algebra, but the author doesn't even bother to give a rigorous proof of the Cauchy-Schwarz inequality (which is super easy) and he instead establishes it by invoking the "cosine law" in an implicitly synthetic framework (thus relying either on axiomatic geometry or lax geometric intuition). This is completely backwards, because my understanding is that you NEED the Cauchy-Schwarz inequality in order to define angles in an Euclidean space (since the inequality tells you that the absolute value of the dot product of two vectors is less than the product of their norms, you can then divide the dot product by the product of the norms and you get a number that is less than 1 in absolute value, and thus you can apply to it the arccos function (whose domain is exactly [-1, 1]) and you DEFINE the angle between the two vectors as this value of the arccos) but instead he invokes the notion of angle in order to prove the Cauchy-Schwarz inequality.

I think I will still finish reading the book since it covers a lot of interesting material (even if many proofs are sketchy) but after I'm done with it I would like to read a more rigorous book that covers similar topics. Can you guys recommend me anything?

[Anonymous No.11353748]

>>11353689
the quran we literary invent sceince an math why ALgebra ALchemy AL come of islam inshallah we hav center of sceicne in turkey audiarabiyya persiyya iraq

[Anonymous No.11353752]

>>11353748
cringe.
>>> /pol/

[Anonymous No.11353754]

>>11353748
Can you take Al Gore back? We don't want him anymore.

[Anonymous No.11353767]

>>11353689
Coxeter, Berger, Dieudonne’s Linear Algebra and Geometry, nigger.

[Anonymous No.11354414]

>>11353689
just write the book yourself