Vector space

In linear algebra, a vector space (also linear space) over the set of scalars $\Lambda$ is a set $\Sigma$ of vectors with two operations:

• Vector addition, a map $+:\Sigma \times \Sigma \to \Sigma$, where the inputs are two vectors $\vec{x},\vec{y}\in \Sigma$ and an output in $\Sigma$. Addition is commutative and associative.
  • Additionally, there exists a neutral element $0\in \Sigma$ such that $x+0=x$.
  • And an additive inverse such that for each $\vec{x}\in \Sigma$ there is some $\vec{y}\in \Sigma$ such that $\vec{x}+\vec{y}=0$.
• Scalar multiplication, a map $\cdot :\Lambda \times \Sigma \to \Sigma$ where it takes a scalar $\alpha \in \Lambda$ and vector $\vec{x}\in \Sigma$ and produces a new vector $\alpha \vec{x}\in \Sigma$. It follows the rule of vector and scalar distributivity and the multiplicative identity element: $1x=x$.

i.e., a vector space is a set of objects that are closed under addition and scalar multiplication. Elements of a vector space are vectors or points.

For instance, ${\mathbb{R}}^n$ is the prototype linear space. An important class of linear spaces are those of functions. For example, if $F(\mathbb{R},\mathbb{R})$ is the set of all functions $\mathbb{R}\to \mathbb{R}$, then $F$ is a linear space if:

$$(f+g)(x)=f(x)+g(x)$$ $$(kf)(x)=kf(x)$$

where the neutral element is $f(x)=0\forall x$. Another broad class of linear spaces is the set of all $n\times m$ matrices ${\mathbb{R}}^{n\times m}$ or the set of complex numbers.

We define subspaces of $\Sigma$ as subsets that contain the neutral element of $\Sigma$, are closed under addition and scalar multiplication.

Subspaces and why these are important

Subspaces are to vector spaces what subsets are to sets. We basically generalise the notion of a subspace with vector spaces.

See also

• Lebesgue space (for applications in signal theory)