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How do linear algebra methods help solve nonlinear problems through approximation?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In numerical analysis, nonlinear problems are frequently solved by reducing them to a sequence of linear problems, a process known as linearization. This approach leverages the efficiency of solving linear systems to approximate solutions for more complex, nonlinear functions.12

A primary example is Newton’s method, which approximates a nonlinear function by its tangent line at a specific point. For a system of nonlinear equations, this method utilizes the Jacobian matrix—a generalization of the derivative—to construct a linear system at each iteration. By solving this linear system, one can find successively better approximations to the root of the original nonlinear system.1

Additionally, functions can be approximated locally using a Taylor series, which represents a function as an infinite sum of terms derived from its derivatives at a single point. By focusing on the linear terms of such expansions, mathematicians can simplify complex behaviors into manageable linear forms, facilitating numerical solutions for problems in optimization and applied mathematics.31