If $f$ : $R^n$ → $R^m$ is a linear map, what would its derivative be?
I believe it's of the form $f(x)$ = $Ax$ w/o the $b$, but either way is ok with me, I just want to know how it works.
derivative of linear mapping
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$\begingroup$
calculus
matrices
derivatives
vector-spaces
vectors
1 Answers
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Suppose that $f(x) = Mx$ for some matrix $M$ (as it must, being a linear map).
The derivative of $f$ at a point $p \in \Bbb R^n$ is a linear transformation $T$; that transformation happens to be $$ T(x) = Mx. $$
If you want the derivative to act on the tangent space to $\Bbb R^n$ at $p$, and map to the tangent space to $\Bbb R^m$ at $f(p)$, then you need coordinates on those tangent spaces. In the natural coordinates induced by the coordinate functions on $\Bbb R^n$ and $\Bbb R^m$, the matrix for $T$ is ... again $M$.