53
$\begingroup$

I'm looking for a book to learn Algebra. The programme is the following. The units marked with a $\star$ are the ones I'm most interested in (in the sense I know nothing about) and those with a $\circ$ are those which I'm mildly comfortable with. The ones that aren't marked shouldn't be of importance. Any important topic inside a unite will be boldfaced.

U1: Vector Algebra. Points in the $n$-dimensional space. Vectors. Scalar product. Norm. Lines and planes. Vectorial product.

$\circ$ U2: Vector Spaces. Definition. Subspaces. Linear independence. Linear combination. Generating systems. Basis. Dimesion. Sum and intersection of subspaces. Direct sum. Spaces with inner products.

$\circ$ U3: Matrices and determinants. Matrix Spaces. Sum and product of matrices. Linear ecuations. Gauss-Jordan elimination. Range. Roché Frobenius Theorem. Determinants. Properties. Determinant of a product. Determinants and inverses.

$\star$ U4: Linear transformations. Definition. Nucleus and image. Monomorphisms, epimorphisms and isomorphisms. Composition of linear transformations. Inverse linear tranforms.

U5: Complex numbers and polynomials. Complex numbers. Operations. Binomial and trigonometric form. De Möivre's Theorem. Solving equations. Polynomials. Degree. Operations. Roots. Remainder theorem. Factorial decomposition. FTA. Lagrange interpolation.

$\star$ U6: Linear transformations and matrices. Matrix of a linear transformation. Matrix of the composition. Matrix of the inverse. Base changes.

$\star$ U7: Eigen values and eigen vectors Eigen values and eigen vectors. Characteristc polynomial. Aplications. Invariant subspaces. Diagonalization.

To let you know, I own a copy of Apostol's Calculus $\mathrm I $ which has some of those topics, precisely:

  • Linear Spaces
  • Linear Transformations and Matrices.

I also have a copy of Apostol's second book of Calc $\mathrm II$which continues with

  • Determinants
  • Eigenvalues and eigenvectors
  • Eigenvalues of operators in Euclidean spaces.

I was reccommended Linear Algebra by Armando Rojo and have Linear Algebra by Carlos Ivorra, which seems quite a good text.

What do you reccomend?

  • 0
    Is this for the CBC, by any chance?2012-06-18
  • 0
    @talmid Precisely. How did you figure?2012-06-18
  • 0
    What is CBC ${}$ ?2012-06-18
  • 0
    @GeorgesElencwajg It is the Basic Common Course (Curso Basico Comun) to all freshmen in the UBA (Univ. of Buenos Aires) - each faculty has a different one, obviously.2012-06-18
  • 1
    I noticed the programme. To pass the CBC, I picked a random book that I found in a bookstore, *Elements of Linear Algebra* by Paige, Swift, and Slobko. I don't think it's very popular, but it covers all this and more. Actually, you'll find all of these topics in just about any book on Linear Algebra. Also, keep in mind that during second year (counting the CBC as first year) you'll study Linear Algebra in more depth, and there's a great book written by some of our teachers specifically for that course. I'd go so far as to recommend you to get that book; it's very rigorous and nicely-written.2012-06-18
  • 1
    Ah, I see. Thank you, Peter, and buena suerte!2012-06-18
  • 0
    @talmid Where are you from?2012-06-18
  • 0
    Why, Buenos Aires, of course! :-) You?2012-06-18
  • 0
    @talmid Same here!2012-06-18
  • 0
    Good luck, then! I think that CBC Algebra isn't very rigorous, and that you'd pass just fine with any book (and some thinking, of course), and then get to learning the real thing during second year. That's what I did since I studied the course during the summer in order to "skip" the CBC altogether; I was in a hurry. But if that's not your situation and you want to make a serious effort now and have the time, you could roll up your sleeves and read something at the level of Hoffman & Kunze, for example, but again, that's not really necessary.2012-06-18
  • 0
    @talmid I was actually thinking of sitting for the "free" exams now in July. It doesn't seem a tough subject.2012-06-18
  • 0
    Great idea. Don't worry too much about learning proofs and theorems in that case. Get some example exams and practice a lot. You'll get back to this subject in more depth and generality later on, and proofs will really matter then. (I want to clarify: I think it's much better to study any subject trying to understand it, thinking a lot about it, and proving theorems and trying to generalize, etc., but when you're hurried by time, there are other things we tend to prioritize.)2012-06-18
  • 0
    @Peter The BIG problem with Hoffman/Kunze is that it's a completely theoretical treatment and the applications of linear algebra are just as important as the underlying theory. In this regard, Friedberg,Insel and Spence's LINEAR ALGEBRA is much more balanced with many applications,while still giving a rigorous presentation of the theory. This is my favorite all purpose LA book.2012-06-18
  • 2
    Gilbert Strang has a nice book, along with free video lectures of the class that it is based on2012-06-19
  • 0
    Lecture notes are usually best in this subject2018-10-28

11 Answers 11

41

"Linear Algebra Done Right" by Sheldon Axler is an excellent book.

  • 1
    I think this is a great book. Plus there are a lot of resources you can use - a course at MIT last fall, check MIT classes last fall for the course number - not the Open courseware. Also Prof. Haiman at Berkeley used this last fall and you can look at his website to get to the course itself. Strang is nice and a lovely persona, but I found his material more of an emphasis on mechanics; whereas Axler is theorem driven.2012-06-18
23

Gilbert Strang has a ton of resources on his webpage, most of which are quite good:

http://www-math.mit.edu/~gs/

  • 2
    Notably, his textbook, videos of his lectures, and class materials are among the resources2012-06-19
  • 0
    this book + videos is all you need.2013-09-22
12

My favorite textbook on the subject by far is Friedberg,Insel and Spence's Linear Algebra, 4th edition. It is very balanced with many applications,including some not found in most LA books,such as applications to stochastic matrices and the matrix exponetioal function,while still giving a comprehensive and rigorous presentation of the theory.It also has many,many exercises-all of which develop both aspects of the subject further. This is without question my favorite all purpose LA book for the serious mathematics student.

7

I think I first learned from Charles W. Curtis' Linear Algebra: An Introductory Approach

Please also note that you will want to use "vector" and "morphism" rather than "vectorial" and "morfism" to get the most hits searching in English.

  • 2
    +1 on a classic more people should be aware of. A very balanced and sophisticated textbook and my second favorite book on the subject.2012-06-18
  • 0
    I'm currently reading this book and am finding it to be very thorough.2012-06-19
1

David Lay's "Linear Algebra and its Applications" is good.

1

S. Winitzki, Linear Algebra via Exterior Products (free book, coordinate-free approach).

0

Carl Meyer Matrix Analysis and Applied Linear Algebra
Beautiful book, modern view point, focuses on how you actually compute the various objects while not sacrificing rigour. Great for self-study.

0

$\underline{Beginner}:$

Ted Shifrin, Linear Algebra: a Geometric Approach
Bernard Kolman, Elementary Linear Algebra with Applications

$\underline{Advance}:$

Hoffman & Kunze, Linear Algebra

-1

Evar Nering's book on linear algebra and matrix theory is also an (old but) excellent textbook. It's free on archive.org.

  • 0
    not with the OP's background2018-02-15
-1

V.V. Voyevodin's textbook on linear algebra is very useful for those studying computational mathematics and theoretical computing.