Difference between revisions of "Jacobian"

I have trouble remembering this but it is trivial to deduce. Notice $$f:\mathbb{R}^m\rightarrow\mathbb{R}^n$$ this means:

(m length vector) $$\mapsto$$ (n length vector)

So for $$x\in \mathbb{R}^m$$ we must have $$Df_x x\in\mathbb{R}^n$$ (in the matrix multiplication sense)

As we go across $$Df_x$$ and down $$x$$ when multiplying, we must have columns (the width of the matrix) = length of the vector, so it is $m$ across, and thus $n$ down.

Furthermore

$$\begin{pmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} & \cdots \\ \vdots & \ddots & \vdots \\ \cdots & \cdots & \cdots \end{pmatrix}\times\begin{pmatrix}x\\y\\ \vdots \end{pmatrix}$$ $$=\begin{pmatrix} \frac{\partial f_1}{\partial x}x+\frac{\partial f_1}{\partial y}y+\cdots \\ \frac{\partial f_2}{\partial x}x+\frac{\partial f_2}{\partial y}y+\cdots \\ \vdots \end{pmatrix}$$ $$=\begin{pmatrix} \text{change in }f_1\text{per }x\text{ times }x +\text{change in }f_1\text{per }y\text{ times }y + \cdots \\ \text{change in }f_2\text{per }x\text{ times }x +\text{change in }f_2\text{per }y\text{ times }y + \cdots \\ \vdots \end{pmatrix}$$

Which makes perfect dimensional sense