Linear transformations and the Jacobian

Matrices as maps, and the Jacobian as the derivative of a vector-valued function.

A matrix defines a linear transformation: it stretches, rotates or projects space. For non-linear maps, the Jacobian matrix plays the same role locally — it generalises the derivative to multivariable, vector-valued functions and describes how the function transforms a small region of space.

J = \frac{\partial f _{1}}{\partial x _{1}} ⋮ \frac{\partial f _{m}}{\partial x _{1}} \dots ⋱ \dots \frac{\partial f _{1}}{\partial x _{n}} ⋮ \frac{\partial f _{m}}{\partial x _{n}}

Example from the lecture: on the unit disk, switching to polar coordinates $x = r cos θ$ , $y = r sin θ$ has Jacobian determinant $r$ , so $\iint sin (x^{2} + y^{2}) d x d y = \int_{0}^{2 π} \int_{0}^{1} sin (r^{2}) r d r d θ$ .

The Jacobian reappears in generative models, where a generator transforms a simple noise distribution into the data distribution.

Formula to remember

Variance

Var (X) = E [(X - μ)^{2}]

Expected squared distance from the mean.

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