Unit 3 Beyond linear maps: local linearization
Unit 1 and Unit 2 focused on linear maps \(\mathbf{x}\mapsto A\mathbf{x}\text{.}\) Most functions are not linear. This unit studies general scalar-valued and vector-valued functions, then uses derivatives and Jacobian matrices (a generalization of derivatives) to recover a linear map that describes a nonlinear function near one point.
\begin{equation*}
F(\mathbf{a}+\mathbf{h})\approx F(\mathbf{a})+J_F(\mathbf{a})\mathbf{h}.
\end{equation*}
Big question. If a function is not linear, what linear map best describes it near one point?
Learning outcomes. By the end of this unit, students should be able to:
Core outcomes.
-
U3-LO1. Classify scalar-valued and vector-valued functions, write component functions, and describe domains, ranges, graphs, and level curves.
-
U3-LO2. Compute limits, continuity checks, derivatives of vector-valued functions, and partial derivatives of multivariable functions as needed.
-
U3-LO3. Compute and interpret partial derivatives, gradients, Hessians, and Jacobian matrices.
-
U3-LO4. Construct tangent planes and local linear approximations for scalar-valued functions.
-
U3-LO5. Use a Jacobian matrix as a local linear map for vector-valued functions: \(F(\mathbf{a}+\mathbf{h})\approx F(\mathbf{a})+J_F(\mathbf{a})\mathbf{h}\text{.}\)
-
U3-LO6. Apply the chain rule in component form and as matrix multiplication of Jacobian matrices.
-
U3-LO7. Interpret simple composed nonlinear systems, such as a tiny neural-network forward pass, by identifying affine pieces, nonlinear pieces, and the local linear approximation.
Toolbox skills. Write component functions, compute partial derivatives and higher partial derivatives, compute gradients and Hessians, compute Jacobian matrices, write tangent plane equations, evaluate local linear approximations, check Jacobian matrix shapes, multiply Jacobian matrices in the chain rule, and compare actual changes with linear predictions.
Application and computation outcomes. Interpret Lab U3
[U3-LO3, U3-LO5, U3-LO6, U3-LO7 | C+R+T | Core] by reading symbolic Jacobian matrices, evaluating a Jacobian matrix at a point, comparing actual nonlinear outputs with local linear predictions, checking Jacobian matrix shapes in the chain rule, reading square-grid diagrams, and identifying affine and nonlinear pieces in small model blocks.
