Local linear approximation states that a continuously differentiable function, whether of one or several variables, behaves like a linear function in a sufficiently small neighborhood of a point, with the discrepancy (error term) vanishing faster than the change in the independent variable(s) — a second-order infinitesimal. This principle, belonging to multivariable calculus, generalizes the single-variable tangent-line approximation (Δf ≈ f′(a)Δx) to the multivariable case (ΔU, ΔV as linear combinations of the partial derivatives times Δx, Δy, or ΔW_lin for n variables), and it motivates the study of systems of linear equations. Matrix algebra is then introduced as the structural framework for representing and combining such linear systems, with matrix coefficients, matrix dimensions (rows/columns), and the row-times-column dot-product rule for matrix multiplication arising directly from the need to compose linear transformations (e.g., chaining variable substitutions analogous to the chain rule).
Local linear approximation states that a continuously differentiable function, whether of one or several variables, behaves like a linear function in a sufficiently small neighborhood of a point, wit…