TITLE: Total derivative of linear transformation
QUESTION [0 upvotes]: I have a task to calculate the total derivatives of the linear transformation $T$ given by its matrix representation 
$$
T = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i\end{pmatrix}.
$$
I know how to calculate total derivatives but I have no idea how to do this.

REPLY [2 votes]: The derivative is the unique linear function $L(\mathbf a): \Bbb R^3 \to \Bbb R^3$ such that $$\lim_{\mathbf h \to 0} \frac{\|T(\mathbf a + \mathbf h) - T(\mathbf a)-[L(\mathbf a)](\mathbf h)\|}{\|\mathbf h\|}=0$$
But, because of the linearity of $T$, we see that $T(\mathbf a+\mathbf h)-T(\mathbf a) = T(\mathbf h)$.  So clearly $L(\mathbf a) = T$.  Meaning the derivative of a linear transformation (at a point $\mathbf a$) is that linear transformation.
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