Let L: U → V be a linear function between inner product spaces. Prove that u ∈ Rⁿ solves the inhomogeneous linear system L[u] = f if and only if Explain why Exercise 3.1.11 is a special case of this result. Remark. Equation (7.83) is known as the weak formulation of the linear system. It plays an essential role in the analysis of differential equations and their numerical approximations, [61].

Data From Exercise 3.1.11

Prove that x ∈ Rⁿ solves the linear system Ax = b if and only if x^TA^Tv = b^Tv for all v ∈ R^m.

The latter is known as the weak formulation of the linear system, and its generalizations are of great importance in the study of differential equations and numerical analysis, [61].

Transcribed Image Text:

(u, L* [v]) = (f,v) for all VEV. (7.83)