The following definitions are used in Exercises 23–30. Definition. If S1 and S2...

The following definitions are used in Exercises 23–30.

Definition. If S1 and S2 are nonempty subsets of a vector space V, then the sum of S₁ and S₂, denoted S₁+S₂, is the set {x+y : x ∈ S₁ and y ∈ S₂}.

Definition. A vector space V is called the direct sum of W₁ and W₂ if W₁ and W₂ are subspaces of V such that W₁ ∩ W₂ = {0} and W₁ +W₂ = V. We denote that V is the direct sum of W₁ and W₂ by writing V = W₁ ⊕ W₂.

Let W be a subspace of a vector space V over a field F. For any v ∈ V the set {v}+W = {v +w: w ∈ W} is called the coset of W containing v. It is customary to denote this coset by v + W rather than {v} + W. (a) Prove that v + W is a subspace of V if and only if v ∈ W. (b) Prove that v ₁ + W = v ₂ + W if and only if v ₁ − v ₂ ∈ W. Addition and scalar multiplication by scalars of F can be defined in the collection S = {v + W: v ∈ V} of all cosets of W as follows:

(v ₁ + W) + (v ₂ + W) = (v ₁ + v ₂ ) + W

for all v ₁ , v ₂ ∈ V and

a(v + W) = av + W

for all v ∈ V and a ∈ F.

(c) Prove that the preceding operations are well defined; that is, show that if v ₁ + W = v ^’ ₁ + W and v ₂ + W = v ^’ ₂ + W, then

(v ₁ + W) + (v ₂ + W) = (v ^’ ₁ + W) + (v ^’ ₂ + W)

And

a(v ₁ + W) = a(v ^’ ₁ + W)

for all a ∈ F.

(d) Prove that the set S is a vector space with the operations defined in (c). This vector space is called the quotient space of V modulo W and is denoted by V/W.