## 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 $T:{\mathbb R}^n\to {\mathbb R}^m$ satisfies linearity (i.e., what the book calls additivity). Suppose also that $T$ is continuous. Show that $T$ is linear (i.e., it also satisfies homogeneity).
b. Give an example of a $T$ 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 $V$ is finite dimensional. What we want to show is that if $V,W$ are vector spaces and $T:V\to W$ is linear, then

$V/{\rm null}(T)\cong {\rm ran}(T)$.

First, we need to make sense of $V/{\rm null}(T)$. Recall that if $A$ is a set, an equivalence relation $\sim$ on $A$ is a relation $\sim\subseteq A\times A$ such that:

• $a\sim a$ for any $a\in A$ (reflexivity),
• Whenever $a\sim b$, then also $b\sim a$ (symmetry),
• If $a\sim b$ and $b\sim c$, then also $a\sim c$ (transitivity).

Given such an equivalence relation, the equivalence class of an element $a\in A$ is the subset ${}[a]\subseteq A$ consisting of all those $b\in A$ such that $a\sim b$. The quotient $A/\sim$ is the collection of all equivalence classes, so if ${\bf x}\in A/\sim$ then there is some $a\in A$ such that ${\bf x}=[a]$.

The point is that the equivalence classes form a partition of $A$ into pairwise disjoint, non-empty sets: Each ${}[a]$ is nonempty, since $a\in[a].$ Clearly, the union of all the classes is $A$ (again, because any $a$ is in the class ${}[a]$), and if ${}[a]\cap[b]\ne\emptyset$, then in fact ${}[a]=[b]$ (check this).

Ok. Back to $T:\to W$. Define, in $V$, an equivalence relation $\sim$ by: $v_1\sim v_2$ iff $Tv_1=Tv_2$ (Check that this is an equivalence relation). Then, as a set, we define $V/{\rm null}(T)$ to be $V/\sim$. The reason why the null space is even mentioned here is because of the following (check this): $v_1\sim v_2$ iff $v_1-v_2\in{\rm null}(T)$.

We want to define addition in $V/{\rm null}(T)$ and scalar multiplication so that $V/\sim$ is actually a vector space.

• Given ${\bf x}$ and ${\bf y}$ in $V/{\rm null}(T)$, set ${\bf x}+{\bf y}={\bf z}$, where if ${\bf x}=[a]$ and ${\bf y}=[b]$, then ${\bf z}=[a+b]$. The problem with this definition is that in general there may be infinitely many $c$ such that ${}[c]=[a]$ and infinitely many $d$ such that ${}[d]=[b]$. In order for this definition to make sense, we need to prove that for any such $c,d$, we ${}[a+b]=[c+d]$. Show this.
• Given ${\bf x}\in V/{\rm null}(T)$, and a scalar $\alpha$, define $\alpha{\bf x}={\bf y}$, where if ${\bf x}=[a]$, then ${\bf y}=[\alpha a]$. As before, we need to check that this is well-defined, i.e., that if ${}[a]=[b]$, then also ${}[\alpha a]=[\alpha b]$.
• Check that $V/{\rm null}(T)$ is indeed a vector space with the operations we just defined.

Now we want to define a linear transformation from $V/{\rm null}(T)$ to ${\rm ran}(T)$, and argue that it is an isomorphism. Define ${\bf T}:V/{\rm null}(T)\to W$ by ${\bf T}({\bf x})=Tv$ where ${\bf x}=[v]$. 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 ${\rm dim}(V/{\rm null}(T))={\rm dim}(V)-{\rm dim}({\rm null}(T))$ if $V$ is finite dimensional. Prove this directly (i.e., without using the statement of Theorem 3.4).

I just want to point out a possible typo. I believe $A \setminus sim$ is supposed to be $A \setminus \sim.$