Linear-system terminology connects linear equations, coefficients, solutions of one equation, and solutions of a system.
For $Ax=b$, $A$ holds the coefficients, $x$ the unknowns, and $b$ the constants. A solution satisfies every row equation; the solution set collects all such vectors. Use parametric form for free variables and parametric vector form to expose the underlying null space.
Scalars are real or complex unless stated otherwise. Over a finite field, a nontrivial affine solution space has finitely many solutions instead of infinitely many.
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- Definition of linear equationDefinition: A linear equation is an equation that can be written in the form $$a 1 x 1 + a 2 x 2 + \cdots + a n x n =...
- Definition of coefficients of a linear equationDefinition: In the linear equation $a 1 x 1 + a 2 x 2 + \cdots + a n x n = b$, the constants $a 1, a 2, \ldots, a n$...
- Definition of solution to a linear equationDefinition: A solution to a linear equation $a 1 x 1 + a 2 x 2 + \cdots + a n x n = b$ is an ordered $n$-tuple of...
- Definition of system of linear equationsDefinition: A system of linear equations (or linear system ) is a finite collection of linear equations involving the...
- Definition of solution to a system of linear equationsDefinition: A solution to a system of linear equations is an ordered $n$-tuple $(s 1, s 2, \ldots, s n)$ that...
- Definition of solution set of a system of linear equationsDefinition: The solution set of a system of linear equations is the set of all solutions to the system. For a system...
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