Linear Equations: Definition and Main Issues
DefinitionA linear equation in Example: We define the solution set to be simply the set of solutions to the linear equation:
Link with subspaces and affine setsRecall the definition of an affine set as the span of a collection of vectors, possibly translated away from the origin. It turns out that solutions sets of linear equations are always affine (Proof). The relationship between linear equations and affine sets goes both ways: any affine set can be expressed as the solution set of some linear equation. Thus, the study of linear equations will help us understand basic geometric concepts such as subspaces and affine sets. In addition, we now know how to parametrize affine sets in two ways: either as the translated span of vectors, or as the solution set of a linear equation Example: hyperplane. Important issuesSeveral important issues arise with linear equations. ExistenceFirst, do solutions exist? This question leads to the notion of range of a matrix, which characterizes those vectors UnicityNext we may be interested in the question of unicity of solutions (if any). The nullspace associated with The notion of nullspace allows to characterize systems for which the solution, if it exists, is unique. If there are many solutions, we can then define the notion of minimum-norm solution. Solution conceptsIf the solution set is empty, that is, the equation does not have any solution, then we may find an approximate solution by solving an optimization problem. If the equation has many solutions to choose from, we can find a particular solution. The one with minimum norm is often of practical interest. SensitivityA linear equation is often obtained from measurement data. In practice, the ‘‘data’’ in the equation (the matrix Sensitivity analysis is concerned with the study of the impact of errors in |