A linear system is consistent when lies in the column space of ; its solutions form a translate of .
A linear equation is an equation in which the unknowns occur only to the first power and are combined linearly.
For a linear map and one solution , all solutions are . This follows because , and conversely adding any kernel element preserves the equation. If is a vector subspace representing a desired regularity class and , then every solution to this one inhomogeneous equation lies in exactly when . For distributional differential equations, take to be the distributions represented by smooth functions.
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A system of linear equations is a collection of two or more linear equations that involve the same set of variables. The goal is to find the values of these variables that satisfy all the equations in the system simultaneously. Systems of linear equations can be classified based on their number of solutions: 1. **Consistent and Independent**: The system has exactly one solution. The lines represented by the equations intersect at a single point.
