Vector space

A vector space over a field (frequently the real numbers) is an object which arises in linear algebra and abstract algebra. A vector space $V$ over a field $F$ consists of a set (of vectors) and two operations, vector addition and scalar multiplication, which obey the following rules:

Axioms of vector space

  • Scalar multiplication is associative, so if $r, s \in F$ and ${\mathbf v} \in V$ then $(rs){\mathbf v} = r(s{\mathbf v})$.
  • Scalar multiplication distributes over vector addition, so if $r \in F$ and $\mathbf{v, w} \in V$ then $r(\mathbf{v + w}) = r{\mathbf v} + r{\mathbf w}$.
  • Scalar multiplication by the multiplicative identity of $F$ is the identity transformation, so $\forall {\mathbf x} \in V$, $1\cdot{\mathbf x}={\mathbf x}$

Subspaces

If $S \subseteq V$ and $S$ is a vector space itself (over the same field), then it is called a subspace of $V$.

Independent Subsets

Let $V$ be any vector space. Let $I$ be a subset of $V$ such that no linear combination of elements of $I$ with coefficients not all zero gives the null vector. Then $I$ is said to be a linearly independent subset of $V$. An independent subset is said to be maximal if on adding any other element it ceases to be independent.

Span

Let $X$ be a subset of some vector space $V$. Then the set of all linear combinations of the elements of $X$ forms a subspace of $V$. This space is said to have been generated by $X$, and is called the span of $X$.

Generating Subset

If $X$ is a subset of a vector space $V$ such that $\textrm{span}(X) = V$, $X$ is said to be a generating subset of $V$. A generating subset is said to be minimal if on removing any element it ceases to be generating.

Basis and dimension

The following statements can be proved using the above definitions:

  • All minimal generating subsets have the same cardinality.
  • All maximal independent subsets have the same cardinality.
  • The cardinality of an independent subset can never exceed that of a generating subset.

An independent generating subset of $V$ is said to be its basis. A basis is always a maximal independent subset and a minimal generating subset. The cardinalities of all bases are equal. This cardinality is said to be the dimension of $V$.

Isomorphism

Any two vector spaces of the same dimension over the same field are isomorphic -- there exists a bijection between the vector spaces which commutes with scalar multiplication and vector addition. Two isomorphic vector spaces are in some sense "the same," and any fact about one should also be true of the other.

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