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).