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Subspaces and Quotient Spaces

Let V be a vector space over the field F. Let us pursue the analogy with the theories of groups and rings and define subspaces and quotient spaces of V.

Definition 1: A subspace of V is a subset of V which is a vector space over F with respect to the operations of vector addition and scalar multiplication of V.

It is clear that W is a subspace of V if and only if W is a subgroup of the additive group of V and W is closed with respect to multiplication by scalars. Thus, by Proposition 2 of the section on subgroups we have

Proposition 2: Let W subset of V. Then W is of subspace of V is equivalent to For all w,w' element of W, alpha element of F, we have w - w' element of W, alphaw element of W.

Example 1: Let V = F2, W = {(a,0) | a element of F}. Then

(a,0) - (a',0) = (a - a',0) element of W,
alpha(a,0) = (alphaa, 0) element of W,

so that by Proposition 2, W is a subspace of V.

Example 2: Let V = C[0,1] (continuous functions on the interval [0,1]), W = {f element of V | f(0) = 0}. Then W is a subspace of V.

Example 3: Let V = F3, W = {(a,b,3a) | a,b element of F}. Then W is a subspace of V.

Let S be an arbitrary subset of V, and let [S] denote the set of all sums of the form

(1)
alpha1s1 + ... + alphamsm,

where alpha1,...,alpham element of F and s1,...,sm element of S. From Proposition 2, it is immediate that [S] is a subspace of V. Moreover, [S] contains S. If W is a subspace of V which contains S, then W contains all the sums of the form (1) and thus W super set of [S]. Therefore, [S] is the smallest subspace of V which contains S. We call [S] the subspace generated by S. If W is a subspace of V and S is a subset of V such that W = [S], then we say that S is a set of generators of W. For example, in Example 1, W = [(1,0)]; in Example 3, W =[S], where S = {(1,0,3),(0,1,0)}. Note that Fn = [S] where S = {e1,...,en} and

e1 = (1,0,0,...,0)
e2 = (0,1,0,...,0)
·
·
·
en = (0,0,0,...,1)

Indeed, if v = (a1,...,an), then

v = a1e1 + a2e2 + ... + anen,

so that Fn subset of [S]. And it is clear that [S] subset of Fn. Thus, Fn = [S].

Let V be a vector space over F and let W be a subspace of V. Then W is a subgroup of the additive group of V. Since the additive group of V is abelian, W is a normal subgroup of V and thus the quotient group V/W is defined. The quotient group V/W consists of the cosets of the form v + W (v element of V) and the group operation is defined by

(v + W) + (v' + W) = (v + v') + W.

If alpha element of F, let us define

(2)
alpha(v + W) = alphav + W.

Let us first show that the above definition is consistent. Indeed, if v + W = v' + W, then v - v' element of W. But since W is a subspace, alpha(v - v') element of W, and thus, since alpha(v - v') = alphav + alpha(-v') = alphav - alphav' we have that

alphav - alphav' element of W implies alphav + W = alphav' + W,
impliesalpha(v + W) = alpha(v' + W).

Thus, the operation (2) is consistent. It is now elementary to check that with respect to the group operation in V/W and scalar multiplication defined by (2), V/W is a vector space over F. The vector space V/W is called the quotient space of V modulo W.

Example 4: Let V = F2, W = {(a,0 | a element of F}. Then since

(a, b) = (a, 0) + (0, b),

we see that every element of V/W can be written in the form

(0, b) + W.

Moreover, no two of these elements are equal, since

(0, b) + W = (0, b') + W implies (0, b - b') element of W
implies(0, b - b') = (a, 0) for some alpha element of F
impliesa = b - b' = 0
impliesb = b'.

Thus, V/W = {(0,b) + W | belement  of F}.

Example 5: Let V = F3, W = {(a,b,0) | a,b element ofF}. Then

V/W = {(0,0,c) + W | c element of F}.