Excerpt from text:
3.109 The range of T'
Suppose V and W are finite-dimensional and T $\in$ L(V,W). Then
range T' = $(null\;T)^0$
Proof
First suppose $\phi$ $\in$ range T. Thus there exists $\psi$ $\in$ W' such that $\phi$ = T'($\psi$). If v $\in$ null T, then
$\phi$(v) = (T'($\psi$))v = ($\psi$ $\circ$ T)(v) = $\psi$(Tv) = $\psi$(0) = 0
Hence $\phi$ $\in$ $(null\;T)^0$. This implies that range T' $\subset$ $(null\;T)^0$.
We will complete the proof by showing that $\mathbf range\;T'$ and $(null\;T)^0$ have the same dimension. To do this, note that
dim range T' = dim range T
dim range T' = dim V – dim null T
dim range T' = dim $(null\;T)^0$
Questions:
I am not able to understand the intuition behind this theorem.
$\mathbf range\; T'$ is the set of functionals from $\mathbf W'$ which take any $\mathbf w$ from $\mathbf W$ to $\mathbb F$ ($\mathbf w$ is produced by $\mathbf T$ here; T'($\phi$) = $\phi \circ T$ for $\phi$ $\in$ W'. $\mathbf Tv$ gives a $\mathbf w$ which is then given to $\phi$() ).
On the other hand, annihilator of $\mathbf (null\;T)$ is the set of all functionals from $\mathbf V'$ which take any vector from the set $\mathbf (null\;T)$ to 0. How can these two different sets be equal?
Would'nt it mean that all functionals from T' are functionals that produce zero?
I am confused about the subset notation in the first part of the proof; " range T' $\subset$ $(null\;T)^0$ " . My current understanding says all functionals in V' which would take vectors from (null T) to zero will be part of range T' . That is, $(null\;T)^0$ $\subset$ range T' . What am I missing here?
(Note: I am assuming $\mathbf (null\;T)$ $\subset$ $\mathbf V$, since annihilator is defined with a subset/superset relation. Have I taken the right superset? )
Also, in the second part of the proof, the author just proves the dimensions of range T' and $(null\;T)^0$ are same. But, how can proving the dimensions being equal prove both spaces are equal and have same elements?
Could anyone kindly explain what I have misunderstood here?
Best Answer
Note that, since $T : V \to W$, we have $T' : W' \to V'$, and hence $\operatorname{range} T'$ is a subset of $V'$, not $W'$. In order for $T'(\psi) = \psi \circ T$ to make sense, the domain of $\psi$ must be from the codomain of $T$, i.e. $W$, not from $V$. The resulting functional therefore has a domain of $V$.
Other than that, I'm not really sure what the source is of your confusion. I usually find people respond well to a worked example, at least to make certain the concepts involved are being understood.
Let's take, for example,
$$T : \mathbb{R}^3 \to \mathbb{R}^2 : (x, y, z) \to (x - y, 3x + y + z).$$
In this case, $V = \mathbb{R}^3$ and $W = \mathbb{R}^2$. Let's start by computing $(\operatorname{null} T)^0$. We have \begin{align*} &(x, y, z) \in \operatorname{null} T \\ \iff \, &T(x, y, z) = (0, 0) \\ \iff \, &(x - y, 3x + y + z) = (0, 0) \\ \iff \, &\exists t \in \mathbb{R} \, : \, (x, y, z) = (t, t, -4t). \end{align*} That is, $$\operatorname{null} T = \lbrace (t, t, -4t) : t \in \mathbb{R} \rbrace = \operatorname{span} (1, 1, -4).$$ Our nullspace is simply this line. As for the annihilator, let us suppose that $\psi(x, y, z) = ax + by + cz$ is a functional in $(\operatorname{span} (1, 1, -4))^0$. That is, $$\psi(1, 1, -4) = 0 \iff a + b - 4z = 0.$$ That is, $$(\operatorname{null} T)^0 = \lbrace (x, y, z) \mapsto ax + by + cz : a + b - 4z = 0 \rbrace.$$ One way to think about functionals is in terms of their nullspaces, all of which are hyperplanes (in this case, just planes, since we are in $\mathbb{R}^3$). The annihilator of $\operatorname{null} T$ is just all of the functionals in $(\mathbb{R}^3)'$ that contain the line $\operatorname{span} (1, 1, -4)$ in their nullspace. Think about all the planes you can form by rotating the plane about this line. These planes represent the functionals in our annihilator.
Now, let's compute $\operatorname{range} T'$. If we take an arbitrary functional $\psi \in (\mathbb{R}^2)'$. Then, there exist $a, b \in \mathbb{R}$ such that $\psi(x, y) = px + qy$. We then have \begin{align*} (T(\psi))(x, y, z) &= (\psi \circ T)(x, y, z) \\ &= \psi(x - y, 3x + y + z) \\ &= p(x - y) + q(3x + y + z) \\ &= (p + 3q)x + (q - p)y + qz. \end{align*} So, the above functional belongs to $(\operatorname{null} T)^0$ if and only if $$(p + 3q) + (q - p) - 4q = 0,$$ which is true, so $\operatorname{range} T' \subset (\operatorname{null} T)^0$.
I think I'll leave it there. The other subset inclusion holds too, but I'll leave you to work it out if you're interested. The theorem is very simple, but the concepts behind them aren't. Hopefully an example has shed some light on these concepts, which might help you grasp the theorem.