Dimension of a quotient space

Theorem 1. If and are complementary subspaces of a vector space , then the correspondence that assigns to each vector in the coset is an isomorphism between and .

Proof. If and are elements of such that , then, in particular, belongs to , so that for some in . Since this means that , and since and are disjoint, it follows that , and hence that . (Recall that belongs to along with and .) 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 , say . Since , we may write in the form , with in and in ; it follows (since ) that . This proves that every coset of can be obtained by using an element of (and not just any old element of ); consequently is indeed a one-to-one correspondence between and . The linear property of the correspondence is immediate from the definition of the linear operations in ; indeed, we have ◻

Theorem 2. If is an -dimensional subspace of an -dimensional vector space , then has dimension .

Proof. Use Section: Dimension of a direct sum , Theorem 2 to find a subspace so that . The space has dimension (by Section: Dimension of a direct sum , Theorem 1), and it is isomorphic to (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 of polynomials modulo various subspaces. If , is finite-dimensional? What if is the subspace consisting of all even polynomials? What if is the subspace consisting of all polynomials divisible by (where )?

Exercise 2. If and are arbitrary subsets of a vector space (not necessarily cosets of a subspace), there is nothing to stop us from defining just as addition was defined for cosets, and, similarly, we may define (where 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 is a subspace of a vector space . Two vectors and of are congruent modulo , in symbols , if is in . Prove that congruence modulo is an equivalence relation , i.e., that it is reflexive ( ), symmetric (if , then ), and transitive (if and , then ).
If and are scalars, and if , , , and are vectors such that and , then .
Congruence modulo splits 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 is an equivalence class modulo if and only if it is a coset of .

Exercise 4.

Suppose that is a subspace of a vector space . Corresponding to every linear functional on (i.e., to every element of ), there is a linear functional on (i.e., an element of ); the linear functional is defined by . Prove that the correspondence is an isomorphism between and .
Suppose that is a subspace of a vector space . Corresponding to every coset of in (i.e., to every element of ), there is a linear functional on (i.e., an element of ); the linear functional is defined by . Prove that is unambiguously determined by the coset (that is, it does not depend on the particular choice of ), and that the correspondence itos an isomorphism between and .

Exercise 5. Given a finite-dimensional vector space , form the direct sum , and prove that the correspondence is an isomorphism between and .

EXERCISES

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