Vector spaces

We come now to the basic concept of this book. For the definition that follows we assume that we are given a particular field $𝔽$ ; the scalars to be used are to be elements of $𝔽$ .

Definition 1. A vector space is a set $𝒱$ of elements called vectors satisfying the following axioms.

(A) To every pair, $x$ and $y$ , of vectors in $𝒱$ there corresponds a vector $x + y$ , called the sum of $x$ and $y$ , in such a way that

addition is commutative, $x + y = y + x$ ,
addition is associative, $x + (y + z) = (x + y) + z$ ,
there exists in $𝒱$ a unique vector $0$ (called the origin ) such that $x + 0 = x$ for every vector $x$ , and
to every vector $x$ in $𝒱$ there corresponds a unique vector $- x$ such that $x + (- x) = 0$ .

(B) To every pair, $α$ and $x$ , where $α$ is a scalar and $x$ is a vector in $𝒱$ , there corresponds a vector $α x$ in $𝒱$ , called the product of $α$ and $x$ , in such a way that

multiplication by scalars is associative, $α (β x) = (α β) x$ , and
$1 x = x$ for every vector $x$ .

(C)

Multiplication by scalars is distributive with respect to vector addition, $α (x + y) = α x + α y$ , and
multiplication by vectors is distributive with respect to scalar addition, $(α + β) x = α x + β x$ .

These axioms are not claimed to be logically independent; they are merely a convenient characterization of the objects we wish to study. The relation between a vector space $𝒱$ and the underlying field $𝔽$ is usually described by saying that $𝒱$ is a vector space over $𝔽$ . If $𝔽$ is the field $ℝ$ of real numbers, $𝒱$ is called a real vector space ; similarly if $𝔽$ is $ℚ$ or if $𝔽$ is $ℂ$ , we speak of rational vector spaces or complex vector spaces .

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