Question

प्रश्न 12. सिद्ध कीजिए कि किसी सदिश समष्टि V (F) के एक अरिक्त
उपसमुच्चय H को V का एक उपसमष्टि होने के लिए आवश्यक एवं पर्याप्त
प्रतिबन्ध है: a,bEF तथा a,BEW Daa+bBEW.

Answer 1

SOLUTION

TO PROVE

A non empty subset W of V(F) is a subspace of V if and only if aα + bβ ∈ W for all α , β ∈ W and a , b ∈ F

PROOF

Let the conditions hold in W

Let α , β ∈ W

aα + bβ ∈ W for all α , β ∈ W and a , b ∈ F

Since - 1 , 0 ∈ F

where 1 is the identity element of F

Then we get - β ∈ W

Since α ∈ W and - β ∈ W

We get α - β ∈ W

∴ α , β ∈ W implies α - β ∈ W

So W is a subgroup of the additive group V

Since V is a commutative group

So W is a commutative subgroup of V

So all conditions for a vector space are satisfied in W

Since W is itself a vector space

So W is a subspace of V

Again the necessity of the conditions follows from the definition of the vector space

Hence proved

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