Theorem 1. If \(\mathcal{M}\) and \(\mathcal{N}\) are complementary subspaces of a vector space \(\mathcal{V}\) , then the correspondence that assigns to each vector \(y\) in \(\mathcal{N}\) the coset \(y + \mathcal{M}\) is an isomorphism between \(\mathcal{N}\) and \(\mathcal{V}/\mathcal{M}\) .
Proof. If \(y_1\) and \(y_2\) are elements of \(\mathcal{N}\) such that \(y_1 + \mathcal{M} = y_2 + \mathcal{M}\) , then, in particular, \(y_1\) belongs to \(y_2 + \mathcal{M}\) , so that \(y_1 = y_2 + x\) for some \(x\) in \(\mathcal{M}\) . Since this means that \(y_1 - y_2 = x\) , and since \(\mathcal{M}\) and \(\mathcal{N}\) are disjoint, it follows that \(x = 0\) , and hence that \(y_1 = y_2\) . (Recall that \(y_1 - y_2\) belongs to \(\mathcal{N}\) along with \(y_1\) and \(y_2\) .) This argument proves that the correspondence we are studying is one-to-one, as far as it goes. To prove that it goes far enough, consider an arbitrary coset of \(\mathcal{M}\) , say \(z + \mathcal{M}\) . Since \(\mathcal{V} = \mathcal{N} + \mathcal{M}\) , we may write \(z\) in the form \(y + x\) , with \(x\) in \(\mathcal{M}\) and \(y\) in \(\mathcal{N}\) ; it follows (since \(x + \mathcal{M} = \mathcal{M}\) ) that \(z + \mathcal{M} = y + \mathcal{M}\) . This proves that every coset of \(\mathcal{M}\) can be obtained by using an element of \(\mathcal{N}\) (and not just any old element of \(\mathcal{V}\) ); consequently \(y \rightarrow y + \mathcal{M}\) is indeed a one-to-one correspondence between \(\mathcal{N}\) and \(\mathcal{V}/\mathcal{M}\) . The linear property of the correspondence is immediate from the definition of the linear operations in \(\mathcal{V}/\mathcal{M}\) ; indeed, we have \[(\alpha_1 y_1 + \alpha_2 y_2) + \mathcal{M} = \alpha_1 (y_1 + \mathcal{M}) + \alpha_2 (y_2 + \mathcal{M}).\] ◻
Theorem 2. If \(\mathcal{M}\) is an \(m\) -dimensional subspace of an \(n\) -dimensional vector space \(\mathcal{V}\) , then \(\mathcal{V}/\mathcal{M}\) has dimension \(n - m\) .
Proof. Use Section: Dimension of a direct sum , Theorem 2 to find a subspace \(\mathcal{N}\) so that \(\mathcal{M} \oplus \mathcal{N} = \mathcal{V}\) . The space \(\mathcal{N}\) has dimension \(n - m\) (by Section: Dimension of a direct sum , Theorem 1), and it is isomorphic to \(\mathcal{V}/\mathcal{M}\) (by Theorem 1 above). ◻
There are more topics in the theory of quotient spaces that we could discuss (such as their relation to dual spaces and annihilators). Since, however, most such topics are hardly more than exercises, involving the use of techniques already at our disposal, we turn instead to some new and non-obvious ways of manufacturing useful vector spaces.
EXERCISES
Exercise 1. Consider the quotient spaces obtained by reducing the space \(\mathcal{P}\) of polynomials modulo various subspaces. If \(\mathcal{M} = \mathcal{P}_n\) , is \(\mathcal{P}/\mathcal{M}\) finite-dimensional? What if \(\mathcal{M}\) is the subspace consisting of all even polynomials? What if \(\mathcal{M}\) is the subspace consisting of all polynomials divisible by \(x_n\) (where \(x_n(t) = t^n\) )?
Exercise 2. If \(\mathcal{S}\) and \(\mathcal{T}\) are arbitrary subsets of a vector space (not necessarily cosets of a subspace), there is nothing to stop us from defining \(\mathcal{S} + \mathcal{T}\) just as addition was defined for cosets, and, similarly, we may define \(\alpha \mathcal{S}\) (where \(\alpha\) is a scalar). If the class of all subsets of a vector space is endowed with these “linear operations,” which of the axioms of a vector space are satisfied?
Exercise 3.
- Suppose that \(\mathcal{M}\) is a subspace of a vector space \(\mathcal{V}\) . Two vectors \(x\) and \(y\) of \(\mathcal{V}\) are congruent modulo \(\mathcal{M}\) , in symbols \(x \equiv y \,\, (\mathcal{M})\) , if \(x - y\) is in \(\mathcal{M}\) . Prove that congruence modulo \(\mathcal{M}\) is an equivalence relation , i.e., that it is reflexive ( \(x \equiv x\) ), symmetric (if \(x \equiv y\) , then \(y \equiv x\) ), and transitive (if \(x \equiv y\) and \(y \equiv z\) , then \(x \equiv z\) ).
- If \(\alpha_1\) and \(\alpha_2\) are scalars, and if \(x_1\) , \(x_2\) , \(y_1\) , and \(y_2\) are vectors such that \(x_1 \equiv y_1 \,\, (\mathcal{M})\) and \(x_2 \equiv y_2 \,\, (\mathcal{M})\) , then \(\alpha_1 x_1 + \alpha_2 x_2 \equiv \alpha_1 y_1 + \alpha_2 y_2 \,\, (\mathcal{M})\) .
- Congruence modulo \(\mathcal{M}\) splits \(\mathcal{V}\) into equivalence classes, i.e., into sets such that two vectors belong to the same set if and only if they are congruent. Prove that a subset of \(\mathcal{V}\) is an equivalence class modulo \(\mathcal{M}\) if and only if it is a coset of \(\mathcal{M}\) .
Exercise 4.
- Suppose that \(\mathcal{M}\) is a subspace of a vector space \(\mathcal{V}\) . Corresponding to every linear functional \(y\) on \(\mathcal{V}/\mathcal{M}\) (i.e., to every element \(y\) of \((\mathcal{V}/\mathcal{M})'\) ), there is a linear functional \(z\) on \(\mathcal{V}\) (i.e., an element of \(\mathcal{V}'\) ); the linear functional \(z\) is defined by \(z(x) = y(x + \mathcal{M})\) . Prove that the correspondence \(y \to z\) is an isomorphism between \((\mathcal{V}/\mathcal{M})'\) and \(\mathcal{M}^0\) .
- Suppose that \(\mathcal{M}\) is a subspace of a vector space \(\mathcal{V}\) . Corresponding to every coset \(y + \mathcal{M}^0\) of \(\mathcal{M}^0\) in \(\mathcal{V}^\prime\) (i.e., to every element \(\mathcal{H}\) of \(\mathcal{V}^\prime/\mathcal{M}^0\) ), there is a linear functional \(z\) on \(\mathcal{M}\) (i.e., an element \(z\) of \(\mathcal{M}'\) ); the linear functional \(z\) is defined by \(z(x) = y(x)\) . Prove that \(z\) is unambiguously determined by the coset \(\mathcal{H}\) (that is, it does not depend on the particular choice of \(y\) ), and that the correspondence \(\mathcal{H} \ z\) itos an isomorphism between \(\mathcal{V}^\prime/\mathcal{M}^0\) and \(\mathcal{M}'\) .
Exercise 5. Given a finite-dimensional vector space \(\mathcal{V}\) , form the direct sum \(\mathcal{W} = \mathcal{V} \oplus \mathcal{V}'\) , and prove that the correspondence \(\langle x, y\rangle \rightarrow \langle y, x \rangle\) is an isomorphism between \(\mathcal{W}\) and \(\mathcal{W}'\) .