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 V. Then W is of subspace of V For all w,w' W, F, we have w - w' W, w W.

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

(a,0) - (a',0) = (a - a',0) W,

(a,0) = (a, 0) 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 V | f(0) = 0}. Then W is a subspace of V.

Example 3: Let V = F³, W = {(a,b,3a) | a,b 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

₁s₁ + ... + _ms_m,

where ₁,...,_m F and s₁,...,s_m 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 [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,...,e_n} and

Indeed, if v = (a₁,...,a_n), then

so that Fⁿ [S]. And it is clear that [S] 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 V) and the group operation is defined by

If F, let us define

(v + W) = v + W.

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

v - v' W v + W = v' + W,

(v + W) = (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 F}. Then since

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

Moreover, no two of these elements are equal, since

(0, b) + W = (0, b') + W (0, b - b') W

(0, b - b') = (a, 0) for some F

a = b - b' = 0

b = b'.

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

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

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