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I have recently been using 3Blue1Brown's fantastic YouTube series on linear algebra to supplement my third year matrix analysis course. I have found it super useful to be able to visualise matrices as transformations of the plane (or 3D space).

As far as I am aware however, he never touches on the effect of taking the transpose of a linear transformation matrix or provide any intuition of what the transpose does to the plane. Could anyone shed light on an intuitive way to think about the transpose of a matrix as a linear transformation.

Architexture
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    [related](https://math.stackexchange.com/questions/37398/what-is-the-geometric-interpretation-of-the-transpose) – Snacc Oct 07 '22 at 16:15
  • If $T : X \rightarrow Y$, then $T^* : Y^*\rightarrow X^*$. The action of $T^*$ is to pull a functional on the range space of $T$ back to a functional on the domain $X$ by composing that functional with $T$. Trying to stick with vector spaces only is not a natural fit for what is going on. – Disintegrating By Parts Oct 08 '22 at 00:33

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