- #1

zenterix

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- Homework Statement
- Theorem 3.107 (Linear Algebra Done Right)

Suppose ##V## and ##W## are finite-dimensional and ##T\in L(V,W)##. Then

(a) null T' = (range ##T)^0##

- Relevant Equations
- I would like to understand item (a) in an intuitive, descriptive way.

There are multiple concepts involved and all are quite abstract. I would like to go over them in my own words before speaking of the theorem in question.

In what follows I do so and end with my understanding of the presented theorem.

Consider the concepts of

Given a linear map ##T\in L(V,W)##, the

Note that given any basis ##v_1, ..., v_n## of ##V##, each distinct linear functional in ##V'## maps the basis vectors ##v_i## in a unique way to scalars in ##\mathbb{F}##.

Consider the ##n## linear functionals defined as

$$\phi_j(v_i)=\begin{cases} 1, \text{ if } i=j \\ 0 \text{ otherwise } \end{cases}$$

for ##j=1,...,n##.

It isn't difficult to see that each linear functional in ##V'## is a linear combination of these ##n## ##\phi_j## linear functionals.

In addition, it can be shown that the ##\phi_j##'s are linearly independent and so are a basis for ##V'##.

We call them the

Given a linear map ##T\in L(V,W)##, the

$$T'(\phi)=\phi\circ T, \text{ for } \phi \in W'\tag{1}$$

Let me try to understand this.

The vectors in the vector space ##L(W', V')## are linear maps that map a linear functional in ##L(W,\mathbb{F})## to a linear functional in ##L(V,\mathbb{F})##.

One specific of these linear maps in ##L(W',V')## is the dual map of ##T\in L(V,W)##.

It is the linear map that maps a ##\phi\in L(W,\mathbb{F})## to ##\phi\circ T##.

##\phi\circ T## is a linear map that maps ##V## to ##W## and then ##W## to ##\mathbb{F}## and thus is in ##V'##.

Next, consider the concept of

Given a vector space ##V## and a subset ##U\subset V##, the annihilator of ##U## is the set of all vectors in ##V'## (ie, linear functionals in ##L(V,\mathbb{F})##) that map ##U## (ie every vector in ##U##) to 0.

The annihilator of ##U## is denoted ##U^0## and is a subspace of ##V'##.

Consider the null space of the dual map ##T'##.

Since ##T'\in L(W',V')## then its null space is a subspace of ##W'##.

##T\in L(V,W)## so the range of ##T## is a subspace of ##W##.

The annihilator of range ##T## is then a subspace of ##W'## (the linear functionals in ##W'## that map range ##T## to 0).

The claim in bold above is that the linear functionals mapped by ##T'## to 0 (that is, the linear functionals ##\phi## such that ##\phi\circ T=0##) are the exact same linear functionals that annihilate the range of ##T##.

Now that I wrote it and thought about it more it finally made clicked and made sense:

Since the dual map is formed by functionals that take a ##v\in V## pass it through ##T## to get a ##w\in \text{range} T## and then pass that through a functional ##\phi## in ##W'##, this can only be zero always if the functional ##\phi## is in the annihilator subspace. Thus, any such ##\phi## in the null space of ##T'## is in the annihilator of range ##T##.

**dual space, dual basis, dual map,**and**annihilator.**Given a linear map ##T\in L(V,W)##, the

**dual space of ##T##**is the vector space ##V'=L(V,\mathbb{F})## where ##\mathbb{F}## is a field.Note that given any basis ##v_1, ..., v_n## of ##V##, each distinct linear functional in ##V'## maps the basis vectors ##v_i## in a unique way to scalars in ##\mathbb{F}##.

Consider the ##n## linear functionals defined as

$$\phi_j(v_i)=\begin{cases} 1, \text{ if } i=j \\ 0 \text{ otherwise } \end{cases}$$

for ##j=1,...,n##.

It isn't difficult to see that each linear functional in ##V'## is a linear combination of these ##n## ##\phi_j## linear functionals.

In addition, it can be shown that the ##\phi_j##'s are linearly independent and so are a basis for ##V'##.

We call them the

**dual basis**of ##v_1, ..., v_n## and they are a basis of ##V'##.**Next let's consider what a dual map is. I find this concept to be the most difficult to grasp so far in linear algebra.**Given a linear map ##T\in L(V,W)##, the

**dual map**of ##T## is a specific linear map in ##L(W',V')##, namely ##T'\in L(W',V')## defined by$$T'(\phi)=\phi\circ T, \text{ for } \phi \in W'\tag{1}$$

Let me try to understand this.

The vectors in the vector space ##L(W', V')## are linear maps that map a linear functional in ##L(W,\mathbb{F})## to a linear functional in ##L(V,\mathbb{F})##.

One specific of these linear maps in ##L(W',V')## is the dual map of ##T\in L(V,W)##.

It is the linear map that maps a ##\phi\in L(W,\mathbb{F})## to ##\phi\circ T##.

##\phi\circ T## is a linear map that maps ##V## to ##W## and then ##W## to ##\mathbb{F}## and thus is in ##V'##.

Next, consider the concept of

**annihilator.**Given a vector space ##V## and a subset ##U\subset V##, the annihilator of ##U## is the set of all vectors in ##V'## (ie, linear functionals in ##L(V,\mathbb{F})##) that map ##U## (ie every vector in ##U##) to 0.

The annihilator of ##U## is denoted ##U^0## and is a subspace of ##V'##.

**Now we get to an extra layer of abstraction.**Consider the null space of the dual map ##T'##.

Since ##T'\in L(W',V')## then its null space is a subspace of ##W'##.

**There is a theorem that says that this null space is the same as the annihilator of the range of ##T##.**##T\in L(V,W)## so the range of ##T## is a subspace of ##W##.

The annihilator of range ##T## is then a subspace of ##W'## (the linear functionals in ##W'## that map range ##T## to 0).

The claim in bold above is that the linear functionals mapped by ##T'## to 0 (that is, the linear functionals ##\phi## such that ##\phi\circ T=0##) are the exact same linear functionals that annihilate the range of ##T##.

Now that I wrote it and thought about it more it finally made clicked and made sense:

Since the dual map is formed by functionals that take a ##v\in V## pass it through ##T## to get a ##w\in \text{range} T## and then pass that through a functional ##\phi## in ##W'##, this can only be zero always if the functional ##\phi## is in the annihilator subspace. Thus, any such ##\phi## in the null space of ##T'## is in the annihilator of range ##T##.

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