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
- 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?