Newton's method (for systems)

GUIDING QUESTION: How can I compute a solution to a non-linear system of equations?

Motivation

Not every problem every problem we'll encounter in practice is linear!

We've developed some good methods for solving linear systems, and we'd like to use them to help us solve non-linear problems.

Motivation

Guiding philosophy: Let's solve a non-linear system by solving a linear system iteratively.

We'll extend the intuition we developed for solving a single non-linear equation in one variable.

Please take a moment to review how we used the tangent line to construct a sequence of approximations to a root of a function $f: \mathbb{R} \to \mathbb{R}$.