Vector space (Linear space)
Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations.
1. Definition
A vector (linear) space over the field $k$ is a set $E$ of elements $x, y, ...$ called vectors with the following algebraic structure:
I. $E$ is an additive abelian group; that is, $E$ is equipped with a mapping denoted by $$E\times E\rightarrow E, (x, y)\mapsto x + y $$
such that the following axioms are satisfied:
(1) $(x+y)+z=x+(y+z)$ (associative law)
(2) $x + y = y + x$ (commutative law)
(3) there exists a vector $0$ such that $x+0= 0+x=x$ for every $x\in E$. Such a vector is called the zero-vector of $E$.
(4) To every vector $x$ there is a vector $-x$ such that $x+(-x)=0$.
II. $E$ is also equipped with a mapping denoted by $$k\times E\rightarrow E, (\lambda, x)\mapsto \lambda x$$
and satisfying the axioms:
(5) $(\lambda\mu)x=\lambda(\mu x)$ (associative law)
(6) $(\lambda+\mu)x=\lambda x+\mu x$
$\lambda (x+ y)=\lambda x+\lambda y$ (distributive laws)
(7) $1 \cdot x = x$ ($1$ unit element of $k$)
$k$ is called the coefficient field of the vector space $E$, and the elements of $k$ are called scalars. Thus the mapping $k\times E\rightarrow E$ defines a multiplication of vectors by scalars, and so it is called scalar multiplication.