Linear algebra, Taylor М., 2020

Linear algebra, Taylor М., 2020.

This text develops linear algebra with the view that it is an important gateway connecting elementary mathematics to more advanced subjects, such as advanced calculus, systems of differential equations, differential geometry, and group representations. The purpose of this book is to provide a treatment of this subject in sufficient depth to prepare the reader to tackle such further material.

Linear algebra, Taylor М., 2020

Rings and modules.
The concept of a ring arose in Chapter 6, on the way to defining the concept of a field. A commutative ring ℛ is endowed with operations of addition and multiplication, satisfying the usual commutative, associative, and distributive laws. One can drop the commutative law of multiplication to obtain the general class of rings. However, most of our attention in this chapter will be devoted to commutative rings with unit, as we discuss the extension of linear algebra from the study of vector spaces over fields to the study of modules over such a class of rings.

Some basic notation.
Chapter 1.Vector spaces, linear transformations, and matrices.
Chapter 2.Eigenvalues, eigenvectors, and generalized eigenvectors.
Chapter 3.Linear algebra on inner product spaces.
Chapter 4.Further basic concepts: duality, convexity, quotients, positivity.
Chapter 5.Multilinear algebra.
Chapter 6.Linear algebra over more general fields.
Chapter 7.Rings and modules.
Chapter 8.Special structures in linear algebra.

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