# What is a good book to study linear algebra?

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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I'm a big fan of Linear Algebra by Hoffman & Kunze. –  Peter Sheldrick Jun 18 '12 at 20:53
We had so many lists of LA books though, it's probably best to link to those. –  Peter Sheldrick Jun 18 '12 at 20:54
Is this for the CBC, by any chance? –  talmid Jun 18 '12 at 20:55
Ah, I see. Thank you, Peter, and buena suerte! –  Georges Elencwajg Jun 18 '12 at 21:07
Gilbert Strang has a nice book, along with free video lectures of the class that it is based on –  David Jun 19 '12 at 2:55

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

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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. –  96 Tears Jun 18 '12 at 23:45

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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G.S. gets my upbote by default ;P –  Peter Sheldrick Jun 18 '12 at 23:28
Notably, his textbook, videos of his lectures, and class materials are among the resources –  David Jun 19 '12 at 2:58
this book + videos is all you need. –  Surya Sep 22 '13 at 14:45

Well, I will just add a few online resources that I have used before,

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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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+1 on a classic more people should be aware of. A very balanced and sophisticated textbook and my second favorite book on the subject. –  Mathemagician1234 Jun 18 '12 at 22:57
I'm currently reading this book and am finding it to be very thorough. –  Andrew Jun 19 '12 at 12:40