Difference between revisions of "Vector space"

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If <math>S \subseteq V</math>, and <math>\mathbf S</math> is a vector space itself, then it is called a subspace of  
 
If <math>S \subseteq V</math>, and <math>\mathbf S</math> is a vector space itself, then it is called a subspace of  
 
<math>\mathbf V</math>.
 
<math>\mathbf V</math>.
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== Independent Subsets ==
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Let <math>\mathbf V</math> be a vector space over the complex field.  Let <math>\mathbf I</math> be a subset of <math>\mathbf V</math> such that no linear combination of elements of <math>\mathbf I</math> with coefficients not all zero gives the null vector. Then <math>\mathbf I</math> is said to be a linearly independent subset of <math>\mathbf V</math>.

Revision as of 14:44, 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 $\mathbf S$ is a vector space itself, then it is called a subspace of $\mathbf V$.

Independent Subsets

Let $\mathbf V$ be a vector space over the complex field. Let $\mathbf I$ be a subset of $\mathbf V$ such that no linear combination of elements of $\mathbf I$ with coefficients not all zero gives the null vector. Then $\mathbf I$ is said to be a linearly independent subset of $\mathbf V$.