Prove that 𝐱 ∈ℝ^n solves the linear system A 𝐱=𝐛 if and only if 𝐱^T A^T 𝐯=𝐛^T 𝐯 for all 𝐯

step 1 In this question we have a matrix A the order of this matrix is m into n we are required to prov

Prove that 𝐱 ∈ℝ^n solves the linear system A 𝐱=𝐛 if and only...

Prove that $\mathbf{x} \in \mathbb{R}^n$ solves the linear system $A \mathbf{x}=\mathbf{b}$ if and only if $$ \mathbf{x}^T A^T \mathbf{v}=\mathbf{b}^T \mathbf{v} \quad \text { for all } \quad \mathbf{v} \in \mathbb{R}^m . $$ 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].

Prove that $\mathbf{x} \in \mathbb{R}^n$ solves the linear system $A \mathbf{x}=\mathbf{b}$ if and only if
$$
\mathbf{x}^T A^T \mathbf{v}=\mathbf{b}^T \mathbf{v} \quad \text { for all } \quad \mathbf{v} \in \mathbb{R}^m .
$$
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].

Key Concepts

Linear Systems

A linear system is a set of linear equations that relate unknown variables through a matrix equation. Solving such a system involves finding a vector that satisfies these equations, typically denoted as Ax = b. This concept underpins many areas of mathematics and applied sciences, and its solutions are essential in computational applications, modeling, and understanding linear transformations.

Weak Formulation

The weak formulation is an alternative way of expressing a linear system, where instead of requiring the equations to hold pointwise, one requires that an integrated or ‘tested’ version of the equation holds against all functions (or vectors) in a certain test space. This approach is crucial in areas such as partial differential equations and finite element methods, as it accommodates solutions that may lack classical differentiability or smoothness and facilitates numerical approximations.

Test Functions (or Test Vectors)

Test functions are functions (or, in finite-dimensional settings, vectors) that are used to 'probe' a system or an equation in the weak formulation. By taking the inner product of the equation with all test functions, one derives conditions that the solution must satisfy in an averaged sense. This idea is central to the weak formulation because it transforms a pointwise condition into one that is integrated against an arbitrary function, thereby making it possible to analyze and compute solutions under broader conditions.

Adjoint Operator

The concept of the adjoint operator, which in the finite-dimensional setting is represented by the matrix transpose, plays an important role in converting the strong (classical) formulation of a linear system into its weak form. By applying the adjoint, one transfers the action from one side of the equation to the other, allowing the equality to be tested against all vectors in the target space. This operation is critical in establishing the equivalence between the classical solution and its weak formulation.

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