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Home » Math Homework Help » Algebra Homework Help » Vector Sub-spaces
Vector Sub-spaces
A subset 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-space

The 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 get



Thus 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

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