# Preliminaries

This chapter contains a variety of definitions of terms that are used throughout the rest of the book, but are not part of linear algebra or do not fit naturally into another chapter. Since these definitions have little connection with each other, a different organization is followed; the definitions are (loosely) alphabetized and each definition is followed by an example.

## Algebra

An (associative) algebra is a vector space A over a field F together with a multiplication (x, y) ↦ xy from A × A to A satisfying two distributive properties and associativity, i.e., for all a, b ∈ F and all x, y, z ∈ A:

$(a\text{x}+b\text{y)z}=a(\text{xy)}+b(\text{yz),}\text{x(}a\text{y}+b\text{z)}=a(\text{xy)}+b(\text{xy)}(\text{xy)z}=\text{x}(\text{yz)}\text{.}$

Except in Chapter 86 and Chapter 87 the term algebra means associative algebra. ...

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