
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 = F^{2}, W = {(a,0)  a F}. Then
(a,0)  (a',0) = (a  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 = F^{3}, 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
(1)
_{1}s_{1} + ... + _{m}s_{m},
where _{1},...,_{m} F and s_{1},...,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 F^{n} = [S] where S = {e_{1},...,e_{n}} and
e_{1} = (1,0,0,...,0)
e_{2} = (0,1,0,...,0)
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·
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e_{n} = (0,0,0,...,1)
Indeed, if v = (a_{1},...,a_{n}), then
v = a_{1}e_{1} + a_{2}e_{2} + ... + a_{n}e_{n},
so that F^{n} [S]. And it is clear that [S] F^{n}. Thus, F^{n} = [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
(v + W) + (v' + W) = (v + v') + W.
If F, let us define
(2)
( 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
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 = F^{2}, W = {(a,0  a 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 (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 = F^{3}, W = {(a,b,0)  a,b F}. Then
V/W = {(0,0,c) + W  c F}.
