403/503- Homework 2

This homework is due February 28.

1. From the textbook: Solve exercises 2.14, 3.3, 3.4, 3.9, 3.10, 3.16, 3.25.

2. a.Suppose  that satisfies linearity (i.e., what the book calls additivity). Suppose also that is continuous. Show that is linear (i.e., it also satisfies homogeneity).
b. Give an example of a that is additive but not homogeneous.

3. The goal of this exercise is to state and prove the rank-nullity theorem (Theorem 3.4 from the book) without the assumption that is finite dimensional. What we want to show is that if are vector spaces and is linear, then

.

First, we need to make sense of . Recall that if is a set, an equivalence relation on is a relation such that:

• for any (reflexivity),
• Whenever , then also (symmetry),
• If and , then also (transitivity).

Given such an equivalence relation, the equivalence class of an element is the subset consisting of all those such that . The quotient is the collection of all equivalence classes, so if then there is some such that .

The point is that the equivalence classes form a partition of into pairwise disjoint, non-empty sets: Each is nonempty, since Clearly, the union of all the classes is (again, because any is in the class ), and if , then in fact (check this).

Ok. Back to . Define, in , an equivalence relation by: iff (Check that this is an equivalence relation). Then, as a set, we define to be . The reason why the null space is even mentioned here is because of the following (check this): iff .

We want to define addition in and scalar multiplication so that is actually a vector space.

• Given and in , set , where if and , then . The problem with this definition is that in general there may be infinitely many such that and infinitely many such that . In order for this definition to make sense, we need to prove that for any such , we . Show this.
• Given , and a scalar , define , where if , then . As before, we need to check that this is well-defined, i.e., that if , then also .
• Check that is indeed a vector space with the operations we just defined.

Now we want to define a linear transformation from to , and argue that it is an isomorphism. Define by where . Once again, check that this is well-defined. Also, check that this is indeed linear, and a bijection.

Finally, to see that this is the “right” version of Theorem 3.4, we want to verify that if is finite dimensional. Prove this directly (i.e., without using the statement of Theorem 3.4).

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