Why, intuitively, is the order reversed when taking the transpose of the product?

It is well known that for invertible matrices A,B of the same size we have (AB)1=B1A1
and a nice way for me to remember this is the following sentence:

The opposite of putting on socks and shoes is taking the shoes off, followed by taking the socks off.

Now, a similar law holds for the transpose, namely:

(AB)T=BTAT

for matrices A,B such that the product AB is defined. My question is: is there any intuitive reason as to why the order of the factors is reversed in this case?

[Note that I’m aware of several proofs of this equality, and a proof is not what I’m after]

Thank you!

Answer

One of my best college math professor always said:

Make a drawing first.

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Although, he couldn’t have made this one on the blackboard.

Attribution
Source : Link , Question Author : user1337 , Answer Author : mdup

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