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:
I also have a copy of Apostol’s second book of Calc $\mathrm II$which continues with
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?
“Linear Algebra Done Right” by Sheldon Axler is an excellent book.
Gilbert Strang has a ton of resources on his webpage, most of which are quite good:
Well, I will just add a few online resources that I have used before,
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.
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.
David Lay’s “Linear Algebra and its Applications” is good.
Evar Nering’s book on linear algebra and matrix theory is also an (old but) excellent textbook.
It’s free on archive.org.