Question

Q1. सदिश समष्टि VF) के एक अरिक्त उपसमुच्चय W को V का एक उपसमष्टि होने के लिए आवश्यक
प्रतिबन्ध है।

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

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

$1.$ A subset B of a vector space V over F is called a basis of V, if:

(A) B is linearly independent set only

(B) B spans...

https://brainly.in/question/30125898

2. Prove that the inverse of the product of two elements of a group is the product of the inverses taken in the reverse ord...

https://brainly.in/question/22739109