# Jacobian Matrix ### Overview The jacobian is fundamentally supposed to represent what a transformation looks like when you zoom in near a specific point. This is taking advantage of local linearity. ### Key Intuitions The jacobian is a matrix that is full of all the partial derivatives of a function. So, if our function was: $f \big( \begin{bmatrix} x \\ y \end{bmatrix} \big) = \begin{bmatrix} x + sin(y) \\ y + sin(x)\end{bmatrix} = \begin{bmatrix} f_1 \\ f_2\end{bmatrix} $ Then the jacobian would be: $\text{Jacobian of } f = Df = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix} $ The Jacobian carries all of the partial differential information of $f$. It takes into account both components of the output, $f_1$ and $f_2$, and both components of the input, $x$ and $y$, giving a "grid" of what the partial derivatives are. We can then *evaluate* the jacobian matrix at a specific point, yielding a matrix of numbers that represents a *linear transformation* at that particular point. ### The Determinant of the Jacobian * The determinant represents how much a linear transformation stretches or squishes space. I.e., how much does it change volume/area? * Evaluating the Jacobian at a specific point and then taking the determinant tells us how much space is being stretched or squished in that region! ### Background Recall that a matrix is a representation of a [linear transformation](Linear%20Transformations.md), which is a transformation of space. We know that we can linear transformations satisfy the property of [Linearity](Linearity.md), which visually means that after the transformation they leave gridlines parallel and evenly spaced:  Nonlinear transformations do not have this property. For example, the nonlinear transformation below transforms space in away that most certainly is not linear:  From this we can see that, in a sense, there is [much more information that goes into nonlinear functions than goes into linear functions](Does%20Linearity%20provide%20Information.md). The above transformation, $f$, still maps from $\mathbb{R}^2 \rightarrow \mathbb{R}^2$, but it seems like it requires far more than 4 numbers to describe where every single point will go (recall, that a 2x2 matrix was all that was required to determine where a linear transformation takes every single point) --- References * [Jacobian prerequisite knowledge](https://www.youtube.com/watch?v=VmfTXVG9S0U&list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7&index=69) * [Local linearity for a multivariable function](https://www.youtube.com/watch?v=Vnga_psnCAo&list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7&index=70) * https://www.youtube.com/watch?v=bohL918kXQk&list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7&index=71 * https://www.youtube.com/watch?v=p46QWyHQE6M&list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7&index=73