Dual of a linear transformation

[Xerxes]

Suppose $V, W$ are vector spaces overthe field $\mathbb{F}$ and that  $L:V\to W$ is a linear transformation.

We know that the dual spaces $V^*, W^*$ exist.

What would be the dual transformation i.e. $V^* \to W^*$? I'm tempted to suggest it has something to do with pullbacks, but I can't seem tp get it to work

Edited March 19 by Xerxes
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[studiot]

  On 3/19/2025 at 6:45 PM, Xerxes said:

Suppose V,W are vector spaces overthe field F and that  L:V→W is a linear transformation.

We know that the dual spaces V∗,W∗ exist.

What would be the dual transformation i.e. V∗→W∗ ? I'm tempted to suggest it has something to do with pullbacks, but I can't seem tp get it to work

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https://math.stackexchange.com/questions/1812391/relation-between-the-dual-space-transpose-matrices-and-rank-nullity-theorem

 

No more time tonight sorry.

[studiot]

  On 3/19/2025 at 6:45 PM, Xerxes said:

I'm tempted to suggest it has something to do with pullbacks, but I can't seem tp get it to work

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Pull-backs are defined for smooth (differentiable) transformations only; for example the pull-back of a Jacobian (which acts on columns from the left) is another matrix of partial derivatives acting on rows from the right.

[Xerxes]

  On 3/19/2025 at 10:45 PM, studiot said:

Pull-backs are defined for smooth (differentiable) transformations only; for example the pull-back of a Jacobian (which acts on columns from the left) is another matrix of partial derivatives acting on rows from the right.

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Well, I believe that the pull-back, like the push-forward, is a more general construction. Here's my thinkinig.....

Suppose the maps (\g:V \to \mathbb{F}\} and (\h:W \to \mathbb{F}\). These are, of course elements of the dual spaces (\V^*\) and (\W^*\}, repectively. Then the pull-back of the map (\f:V \to W\) maps (\W^* \) onto (\V^*\),which is not quite what I was looking for

Edited March 20 by Xerxes