Lesson 34 of 48
Jacobians and vector fields
Derivatives when both the input and the output have many dimensions.
The Jacobian is the local linear approximation of a multi-in, multi-out map — the matrix that says what the function looks like if you stand close enough. Chained Jacobians are literally what backprop multiplies.
Do this
Compute the Jacobian of a map from two dimensions to two by hand and verify every entry numerically. Then push a small square near a point through the map and compare it to the Jacobian's prediction.
The question that unlocks the next lesson
For a map from R^3 to R^2, the Jacobian has what shape?
- A3×3
- B2×3
- C3×2
- D2×2