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Wikipedia says that given $f(x) = x^TA$: $$\frac{\partial f}{\partial x} = A^T$$

but I am having trouble understanding this result. I tried doing the following:

$$f(x+h) - f(x) = h^TA$$

I am now stuck at making the above expression into a linear mapping of $h$. I'm not sure how to relate this to $A^T$. If $A$ were a vector, it would be easy because I could just switch the transpose and have $h^Ta = a^Th$. Since $A$ is a matrix, I am not sure how to proceed.

Rainroad
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  • Does writing the function as $,f=A^Tx,$ help with your understanding? – greg Oct 31 '21 at 03:33
  • isn't that the transpose of f? – Rainroad Oct 31 '21 at 04:20
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    There is no essential difference between $x^TA$ and $A^Tx$, they are just the same vector represented as a row or a column. Usually you choose either a column or a row representation and stick to it. But in this wikipedia page $f(x)=x^TA$ maps a column vector to a row vector, and hence the confusion. Apparently it is not very self-contained. – trisct Oct 31 '21 at 05:21
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    I'll also add that the transpose is a linear function so that it is its own derivative. That allows you to calculate this using the chain rule. – Daniel Oct 31 '21 at 17:55

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