Transpose of a linear map

Okay, so imagine you have a big box of legos. You can build lots of things with the legos, like a castle or a car or a robot. Now, a linear map is like a set of instructions for how to build something with those legos. And the transpose of a linear map is a way to look at those instructions in a different way.

Let's say you have a set of instructions for how to build a castle with the legos. The instructions might say things like "put the blue lego on top of the red lego" or "stack four yellow legos on top of each other". Now, the transpose of those instructions would be like flipping them upside down and backwards. So instead of "put the blue lego on top of the red lego", it would be "put the red lego on the bottom and the blue lego on top".

Why would we want to do this? Well, sometimes it can be easier to build things in a different way. Maybe it's easier to put the blue lego on top of the red lego if you do it in a certain order. And the transpose of the instructions tells you what that order is.

In math, we use the transpose of a linear map to do things like finding eigenvectors or solving systems of equations. It's just a different way of looking at those instructions for how to build something with legos.