The Jacobian essentially represents what a multivariable function looks like locally (when zooming in on a specific point), as a linear transformation. ![[locally_linear.png]] When taking a point $\mathbf{v}$ to be transformed by the (non-)linear transformation $F$ as $\mathbf{u} = F \mathbf{v}$ where $ F = \begin{bmatrix} f_1(x,y) \\ f_2(x,y) \end{bmatrix} \text{,} $ the change in the **output space** in $x$ is the change of $f_1$ w.r.t $x$ **plus** the change of $f_1$ w.r.t. $y$, and the change in $y$ (still in the **output space**) is the change of $f_2$ w.r.t. $x$ **plus** the change of $f_2$ w.r.t. $y$. As such, the matrix describing these changes in $x$ and $y$ is the Jacobian matrix, expressed as: $ \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}. $ This matrix defines the new basis vectors for *a linear transformation*. It is *linear* because partial derivatives describe infinitesimally small changes in the input, which is like zooming in on the a point in the non-linear transformation. > [!hint] Takeaway > When evaluated at a point, the Jacobian matrix shows what the linear transformation looks like locally. In *linear algebra*, the Jacobian is the matrix of first-order partial derivatives for vector-valued functions.