Theorem-Vector Space

Statement:-  

The necessary & sufficient conditions for a non-empty subset W of a vector spaceV(F) to be a subspace of V are

( 1)  α∊W, β∊W ⇨ α-β∊W

(2)  a∊F, α∊W ⇨ aα∊W

Proof:-                Necessary Condition:-

Let V be a vector space over the field F and W be its subspace.

∴  W will be a vector space over the same field F.

Also,

          (W,+) is an abelian group.

∴   If   β∊W  ⇒  -β∊W

          α∊W, -β∊W  ⇒  α+(-β)∊W (By inverse axiom)

   ⇒  α-β∊W

Hence,

α∊W, β∊W  ⇒  α-β∊W ,∀ α,β∊W

Also,

          W will be closed under scalar multiplication.

∴   If a∊F, α∊W ⇒ aα∊W , ∀ a∊F, α∊W

                   Sufficient Condition:-

Let W be a non-empty subset of a vector space V(F) satisfying the condition-

1- α∊W, β∊W  ⇒ α-β∊W, ∀ α,β∊W

2- a∊F,  α∊W  ⇒ aα∊W, ∀ a∊F, α∊W

∴     from(1)

          α∊W, β∊W  ⇒  α-β∊W

⇒      α∊W, α∊W  ⇒  α-α∊W

i.e.   0∊W

∴    The zero vector of V also exists inW.

          0∊W, β∊W  ⇒  0-β∊W

⇒           -β∊W

∴        β∊W  ⇒  -β∊W,  ∀ β∊W

Hence,  each element of W have their additive inverse.

∵            α∊W,  -β∊W

⇒          α-(-β)∊W

⇒            α+β∊W

W is closed under vector addition.

∴   from(2)

          a∊F, α∊W  ⇒  aα∊W, ∀ a∊F,  α∊W

i.e.  W is closed under scalar multiplication.

Since,

          All the elements of W are the elements of V. So associativity and community of vector addition are satisfied by elements of W.