228 Linear Space
A set of vectors together with addition of vectors and multiplication by scalars that is used to build domains for functions and for linear transformations.
Note: Also called vector space.
definition [d] (Linear Space = Vector Space) From Kreyszig Definition 2.1-1: a vector space over a field \(K\) is a nonempty set \(X\) of elements \(x, y, \ldots\) called vectors together with two algebraic operations called vector addition and multiplication of vectors by scalars, that is, by elements of \(K\).
Vector addition associates with every ordered pair \((x,y)\) of vectors a vector \(x+y\) such that vector addition is commutative and associative. Multiplication by scalars satisfies
- \(\alpha(\beta x) = (\alpha\beta)x\)
- \(1x = x\)
and the distributive laws
- \(\alpha(x+y) = \alpha x + \alpha y\)
- \((\alpha+\beta)x = \alpha x + \beta x\) .
Vector addition is a mapping \(X \times X \rightarrow X\), and multiplication by scalars is a mapping \(K \times X \rightarrow X\).
where
- \(X\) is the set of vectors.
- \(K\) is the scalar field.
- \(X\) is a real vector space if \(K = \mathbb{R}\).
- \(X\) is a complex vector space if \(K = \mathbb{C}\).
228.1 Elementary Example
228.1.1 Simple
A linear space has vector addition and scalar multiplication. On three labeled vectors in \(\mathbb{R}\), addition is ordinary.
\[ X = \{ 0,\ 1,\ -1 \} \]
\[ 1 + (-1) = 0,\quad 2 \cdot 1 = 2\ \text{in }\mathbb{R} \]
where
- \(X\) illustrates elements of a real vector space.
- the ambient operations are those of \(\mathbb{R}\) as a vector space over itself.