- #1

zenterix

- 678

- 83

- Homework Statement
- Prove the following theorem (Axler, Linear Algebra Done Right, Theorem 3.109)

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

(a) ##\text{dim range}\ T' = \text{dim range}\ T##

(b) ##\text{range}\ T'=(\text{null}\ T)^0##

- Relevant Equations
- (a)

##\text{dim null}\ T' = \text{dim null}\ T + \text{dim}\ W-\text{dim}\ V\tag{1}##

##=\text{dim} W'-\text{dim range}\ T'\tag{2}##

##=\text{dim}\ W-\text{dim range}\ T'\tag{3}##

From (1) and (3) we have

##\text{dim range}\ T'=\text{dim}\ V-\text{dim null}\ T=\text{dim range}\ T\tag{4}##

My question is about item (b).

(b)

Here is what I drew up to try to visualize the result to be proved

The general idea, I think, is that

1) ##(\text{null}\ T)^0## and ##\text{range}\ T'## are both subspaces of ##V'=L(V,\mathbb{F})##.

2) We can show that they have the same dimension.

3)We can show that ##\text{range}\ T' \subseteq (\text{null}\ T)^0##

4) From (2) and (3) we can prove that the two subspaces are in fact the same subspace.

Honestly, I did steps (1)-(3) myself and was looking for a way to infer (4) but didn't realize that I could use (2) and (3) to do so, so I looked at the proof in the book.

I'd like to know if there is another way to infer (4).

Here is what I have

##(\text{null}\ T)^0## is by definition all the linear functionals in ##V'## that map ##\text{null}\ T## to 0 in ##\mathbb{F}##.

For every ##\varphi\in W'##, the linear functional ##\varphi\circ T## maps ##\text{null}\ T## to 0 in ##\mathbb{F}##, so ##\varphi\circ T## is in ##(\text{null}\ T)^0##.

Now, at this point, it could be that there are other elements in ##(\text{null}\ T)^0## that are not one of the ##\varphi\circ T##.

How can I prove that this is not possible (in an alternative manner to (4))?

(b)

Here is what I drew up to try to visualize the result to be proved

The general idea, I think, is that

1) ##(\text{null}\ T)^0## and ##\text{range}\ T'## are both subspaces of ##V'=L(V,\mathbb{F})##.

2) We can show that they have the same dimension.

3)We can show that ##\text{range}\ T' \subseteq (\text{null}\ T)^0##

4) From (2) and (3) we can prove that the two subspaces are in fact the same subspace.

Honestly, I did steps (1)-(3) myself and was looking for a way to infer (4) but didn't realize that I could use (2) and (3) to do so, so I looked at the proof in the book.

I'd like to know if there is another way to infer (4).

Here is what I have

##(\text{null}\ T)^0## is by definition all the linear functionals in ##V'## that map ##\text{null}\ T## to 0 in ##\mathbb{F}##.

For every ##\varphi\in W'##, the linear functional ##\varphi\circ T## maps ##\text{null}\ T## to 0 in ##\mathbb{F}##, so ##\varphi\circ T## is in ##(\text{null}\ T)^0##.

Now, at this point, it could be that there are other elements in ##(\text{null}\ T)^0## that are not one of the ##\varphi\circ T##.

How can I prove that this is not possible (in an alternative manner to (4))?

Last edited: