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Vector Sub-spaces

A subset

Criterion for a subset W of a vector space

The necessary and sufficient conditions for a subset

Thus the zero vector of

the additive inverse of each element of

Thus

Since the elements of

**W**of a vector space**V(K)**is said to be vector subspace or simply a sub-space of**V(K)**if it is stable for the two compositions defined in**V(K)**and is a vector space for the induced compositions.Criterion for a subset W of a vector space

**V(K)**to be a sub-spaceThe necessary and sufficient conditions for a subset

**W**of vector space**V**over**K**, i.e.**V(K)**to be a subspace are that**Proof:**The conditions necessary: Let**W**be a subspace of**V W**is an abelian group with respect to vector addition. Thus Also**W**must be closed under scalar multiplication. Therefore, the condition**(ii)**is also true.**The conditions are sufficient:**Suppose**W**is a non-empty subset of**V**satisfying the two given conditions. From condition**(i)**, we getThus the zero vector of

**V**belongs to**W**and it will also be the zero vector of**W**.the additive inverse of each element of

**W**is also in**W**.Thus

**W**is closed with respect to vector addition.Since the elements of

**W**are also the elements of**V**, therefore, vector addition is commutative as well as associative in**W**. Hence**W**is an abelian group under vector addition. Also from condition**(ii)**,**W**is closed under scalar multiplication. The remaining postulates of a vector space will hold in**W**since they hold in**V**which is a superset of**W**. Hence**W**is a subspace of**V**.**Services: -**Vector Sub-spaces Homework | Vector Sub-spaces Homework Help | Vector Sub-spaces Homework Help Services | Live Vector Sub-spaces Homework Help | Vector Sub-spaces Homework Tutors | Online Vector Sub-spaces Homework Help | Vector Sub-spaces Tutors | Online Vector Sub-spaces Tutors | Vector Sub-spaces Homework Services | Vector Sub-spacesSubmit Your Query ???

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