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 19Mar 19 by Xerxes
latex

Just now, 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

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

 

No more time tonight sorry.

Just now, Xerxes said:

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

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.

19 hours ago, 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.

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 20Mar 20 by Xerxes