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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?

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  • $\begingroup$ Is this for the CBC, by any chance? $\endgroup$
    – talmid
    Commented Jun 18, 2012 at 20:55
  • $\begingroup$ @talmid Precisely. How did you figure? $\endgroup$
    – Pedro ♦
    Commented Jun 18, 2012 at 20:58
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    $\begingroup$ 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. $\endgroup$
    – talmid
    Commented Jun 18, 2012 at 21:06
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    $\begingroup$ Ah, I see. Thank you, Peter, and buena suerte! $\endgroup$
    – Georges Elencwajg
    Commented Jun 18, 2012 at 21:07
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    $\begingroup$ Gilbert Strang has a nice book, along with free video lectures of the class that it is based on $\endgroup$
    – David LeBauer
    Commented Jun 19, 2012 at 2:55

12 Answers 12

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

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    $\begingroup$ 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. $\endgroup$
    – user12802
    Commented Jun 18, 2012 at 23:45
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Gilbert Strang has a ton of resources on his webpage, most of which are quite good:

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

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    $\begingroup$ Notably, his textbook, videos of his lectures, and class materials are among the resources $\endgroup$
    – David LeBauer
    Commented Jun 19, 2012 at 2:58
  • $\begingroup$ this book + videos is all you need. $\endgroup$
    – Surya
    Commented Sep 22, 2013 at 14:45
  • $\begingroup$ While this book has a lot of resources, I dislike the author approach. He is using things without properly defining them. Already on the 1st page the word "determinant" pops up out of nothing. The same story about the term "singular matrix"--exact definition is nowhere to be found. The author sometimes starts describing the methods, for instance LU decomposition, but then stops without providing crucial details. That is the way a student would write a book, not a professional mathematician. $\endgroup$
    – yarchik
    Commented Oct 27, 2022 at 7:48
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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.

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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.

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    $\begingroup$ +1 on a classic more people should be aware of. A very balanced and sophisticated textbook and my second favorite book on the subject. $\endgroup$
    – Mathemagician1234
    Commented Jun 18, 2012 at 22:57
  • $\begingroup$ I'm currently reading this book and am finding it to be very thorough. $\endgroup$
    – Andrew
    Commented Jun 19, 2012 at 12:40
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David Lay's "Linear Algebra and its Applications" is good.

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$\underline{Beginner}:$

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

$\underline{Advance}:$

Hoffman & Kunze, Linear Algebra

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

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  • $\begingroup$ not with the OP's background $\endgroup$
    – TheLast Cipher
    Commented Feb 15, 2018 at 7:13
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S. Winitzki, Linear Algebra via Exterior Products (free book, coordinate-free approach).

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

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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.

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Serge Lang has an Introduction to Linear Algebra, and a more advanced $\textit{Linear Algebra}$. Both of the pdfs can be found easily (on google) for free. Lang is very good at getting the point across and synthesizing abstract concepts with lots of examples. They are both more or less undergraduate level. The introduction book is more-so intended for a gentle engineering course, while the other book focuses on efficiently developing the structure theorems for vector spaces, including the spectral theorem, symmetric diagonalization, triangulation, invariant subspaces, and the Jordan Normal Form. Lang's program in the second book is extremely well laid-out, but he was an extremely experienced and talented writer, so his first book is a good choice also, if that is more of what you want.

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