# Good books on linear algebra and real/complex analysis?

• Analysis
Hey everyone! (new to the forum)

I am currently trying to self study more advanced mathematics. I have taken up to multivariable calculus and have taken a class for an introduction to mathematical proofs/logic (sets, relations, functions, cardinality). I want to get a head start on the abstract math that would help me get a deeper understanding of mathematical tools used in physics. Could anyone, maybe, give me some recommendations on a linear algebra, real/complex analysis book(s)? My hope is to eventually get an understanding of topology, differential geometry and tensor calculus. Next year I am going to take a two semester sequence in abstract algebra so I won't need any self learning of that (at least I hope not).

I hope that is enough information.

Thanks!
Tom

mathwonk
Homework Helper
2020 Award
here is a good free linear algebra book:

here are my own linear algebra notes, NOWHERE NEAR as good as the previous link, but since i wrote them, i hope someday someone will read them and give me some feedback.

http://alpha.math.uga.edu/~roy/laprimexp.pdf

my favorite complex intro is by Frederick P. Greenleaf, Intro to complex variables.

https://www.abebooks.com/book-searc...complex-variables/author/frederick-greenleaf/

It is harder to think of an excellent intro to reals, maybe something by George Simmons.

http://susanka.org/HSforQM/[Simmons]_Introduction_to_Topology_and_Modern_Analysis.pdf

these are chosen for their accessibility to the average student. if you are a super advanced student you may want more advabced stuff, which i am also glad to suggest. but these include ones i myself could learn from as a young student.

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IMHO, it's relatively easy to get through a linear algebra course and end up with good computation skills but relatively weak intuition for what it all means.

A great supplement to mathwonk's recommendation is this short youtube series whose goal is to flesh out that intuition:

I second ibkev's recommendation of 3Blue1Brown's YouTube series.

If you want a physical book to learn the basics of linear algebra from, the best bang for the buck I've found so far is used copies of David Lay's Linear Algebra and Its Applications, 3rd Updated Edition. If you're in the USA, they're easy to find for very cheap on Amazon.

I learned complex analysis in a very circuitous way over several years through many books, so I don't have a good recommendation there. I think it would be easy enough to present the basics of what physicists learn of complex analysis in undergrad but pitched towards undergraduate mathematicians at a similar level to a class on vector calculus, but I don't personally know of such a text.

Once you have the basics of complex analysis, though, Tristan Needham's book Visual Complex Analysis is perfect for building visual and geometric intuition with the subject. I recommend it in a similar way to the abovementioned video series by 3Blue1Brown. 3Blue1Brown has some nice videos on complex numbers, too, but they don't go into the same depth as Needham's book. I recommend you watch them, but they aren't very comprehensive on this topic.

I really enjoyed Serge Lang: Introduction to Linear Algebra. Really great and concise book. It lacks problems, so maybe supplement it with Anton or another run of the mill book.

mathwonk