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zenterix
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- Homework Statement
- Suppose ##V## and ##W## are finite-dimensional, ##T\in L(V,W)##, and there exists ##\phi\in W'## such that null ##T'## = span(##\phi##). Prove that range ##T## = null ##\phi##.
- Relevant Equations
- null ##T'## = span(##\phi##)
This is problem 28 from chapter 3F "Duality" of Axler's Linear Algebra Done Right, third edition.
I spent quite a long time on this problem, like a few hours. Since there is no available solution, I am wondering if my solution is correct.
One assumption in this problem is that ##\text{null}(T')=\text{span}(\phi)## where ##\phi\in W'##
We know from a previous theorem that
$$\text{null}(T')=(\text{range}(T))^0\tag{2}$$
The reason for this is as follows
##\text{null}(T')## is formed by ##\phi\in W'## such that
$$\phi(\text{range}(T))=0\tag{3}$$.
Thus, $$\text{null}(T')=\{\phi\in W':\phi v=0\ \forall v\in\ \text{range}\ T\}=(\text{range}(T))^0\tag{4}$$
Moving on, since a single non-zero vector is linearly independent (l.i.) then ##\phi## is a basis for ##\text{null}(T')##. All other ##\alpha\in\ \text{null}(T')## are a multiple of ##\phi##.
Note that for each such ##\alpha\in\ \text{null}(T')##, (4) tells us that every element in ##\text{range}\ T## is in ##\text{null}\ \phi##. That is, $$\text{range}(T) \subseteq\ \text{null} (\phi)\tag{5}$$
Claim: all ##\alpha\in\ \text{null}(T')## have the same nullspace.
Suppose, for proof by contradiction, that ##\text{range}(T) \neq\ \text{null}(\phi)##. That is, there is some ##w\in\ \text{null}(\phi)## that is not in range##(T)##.
Then, ##\phi(w)=0##, which means that not only does ##\phi## annihilate range##(T)##, it also annihilates the subspace span##(w)##.
Now, as we proved above, all ##\alpha\in\ \text{null}(T')## have the same nullspace and thus
$$\forall\ \alpha\in\ \text{null}(T) \implies\ \alpha w=0\tag{9}$$
Consider a ##\beta\in W'## such that ##\forall\ x\in\ W## we have
$$\beta x = \begin{cases} \phi x,\ \text{if}\ x\in\ \text{range}(T) \\ 1,\ \text{if}\ x\in\ W \backslash\text{range}(T) \end{cases}\tag{10}$$
Then, ##\beta\in\ (\text{range}(T))^0=\text{null}(T')##.
But ##w\notin\ \text{null}(\beta)##, which contradicts the fact that all linear functionals in ##\text{null}(T')## have the same nullspace which includes ##w## since by assumption ##\phi (w)=0##.
Thus, but proof by contradiction, we infer that $$\text{range}\ T =\ \text{null}\phi$$.
I spent quite a long time on this problem, like a few hours. Since there is no available solution, I am wondering if my solution is correct.
One assumption in this problem is that ##\text{null}(T')=\text{span}(\phi)## where ##\phi\in W'##
We know from a previous theorem that
$$\text{null}(T')=(\text{range}(T))^0\tag{2}$$
The reason for this is as follows
##\text{null}(T')## is formed by ##\phi\in W'## such that
$$\phi(\text{range}(T))=0\tag{3}$$.
Thus, $$\text{null}(T')=\{\phi\in W':\phi v=0\ \forall v\in\ \text{range}\ T\}=(\text{range}(T))^0\tag{4}$$
Moving on, since a single non-zero vector is linearly independent (l.i.) then ##\phi## is a basis for ##\text{null}(T')##. All other ##\alpha\in\ \text{null}(T')## are a multiple of ##\phi##.
Note that for each such ##\alpha\in\ \text{null}(T')##, (4) tells us that every element in ##\text{range}\ T## is in ##\text{null}\ \phi##. That is, $$\text{range}(T) \subseteq\ \text{null} (\phi)\tag{5}$$
Claim: all ##\alpha\in\ \text{null}(T')## have the same nullspace.
Proof
By assumption, ##\text{null}(T')=\ \text{span}(\phi)##.
Then, ##\forall\ \alpha\in\ \text{null}\ T'\implies \alpha=\lambda\phi,\ \lambda\in\mathbb{F}##.
Let ##w\in\ \text{null}(\alpha)##. Then
$$\alpha (w) = (\lambda\phi)(w) = 0\implies \phi(w)=0\tag{6}$$
$$w\in\ \text{null}(\phi)\tag{7}$$
Now, suppose that ##w\in \text{null}(\phi)##. Then
$$\phi(w)=0 \implies \alpha(w) = (\lambda \phi)(w) = 0$$
$$w\in\ \text{null}(\alpha)\tag{8}$$
Therefore, we can infer that ##\text{null}(\alpha)=\text{null}(\phi)##, and since this is true for a generic ##\alpha\in\ \text{null}(T')## then it is true for them all.
Suppose, for proof by contradiction, that ##\text{range}(T) \neq\ \text{null}(\phi)##. That is, there is some ##w\in\ \text{null}(\phi)## that is not in range##(T)##.
Then, ##\phi(w)=0##, which means that not only does ##\phi## annihilate range##(T)##, it also annihilates the subspace span##(w)##.
Now, as we proved above, all ##\alpha\in\ \text{null}(T')## have the same nullspace and thus
$$\forall\ \alpha\in\ \text{null}(T) \implies\ \alpha w=0\tag{9}$$
Consider a ##\beta\in W'## such that ##\forall\ x\in\ W## we have
$$\beta x = \begin{cases} \phi x,\ \text{if}\ x\in\ \text{range}(T) \\ 1,\ \text{if}\ x\in\ W \backslash\text{range}(T) \end{cases}\tag{10}$$
Then, ##\beta\in\ (\text{range}(T))^0=\text{null}(T')##.
But ##w\notin\ \text{null}(\beta)##, which contradicts the fact that all linear functionals in ##\text{null}(T')## have the same nullspace which includes ##w## since by assumption ##\phi (w)=0##.
Thus, but proof by contradiction, we infer that $$\text{range}\ T =\ \text{null}\phi$$.
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