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Difference between revisions of "Vector space"
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Let <math>V</math> be a vector space over the complex field.  Let <math>I</math> be a subset of <math>V</math> such that no linear combination of elements of <math>I</math> with coefficients not all zero gives the null vector. Then <math>I</math> is said to be a linearly independent subset of <math>V</math>.
 
Let <math>V</math> be a vector space over the complex field.  Let <math>I</math> be a subset of <math>V</math> such that no linear combination of elements of <math>I</math> with coefficients not all zero gives the null vector. Then <math>I</math> is said to be a linearly independent subset of <math>V</math>.
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== Linear Manifolds ==
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Let <math>X</math> be a subset of some vector space <math>V</math>. Then it can be proved that the set of all linear combinations of the elements of <math>X</math> forms a vector space. This space is said to have been generated by <math>X</math>, and is called the linear manifold <math>M(X)</math> of <math>X</math>.

Revision as of 13:55, 4 November 2006

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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 a 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 is distributive over both vector and scalar addition, so if $r \in F$ and ${\mathbf v, w} \in V$ then $r({\mathbf v + w}) = r{\mathbf v} + r{\mathbf w}$.
  • if $x \in V$, $1.{\mathbf x}={\mathbf x}$

Subspaces

If $S \subseteq V$, and $S$ is a vector space itself, then it is called a subspace of $V$.

Independent Subsets

Let $V$ be a vector space over the complex field. 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 $V$.

Linear Manifolds

Let $X$ be a subset of some vector space $V$. Then it can be proved that the set of all linear combinations of the elements of $X$ forms a vector space. This space is said to have been generated by $X$, and is called the linear manifold $M(X)$ of $X$.

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