A linear system is a finite list of equations in the same unknowns, written $Ax=b$ with $A\in\mathbb F^{m\times n}$. A solution must satisfy every equation. The system is consistent iff $b\in\operatorname{Col}A$. If consistent, its solution set is $x_p+\ker A$, an affine space of dimension $n-\operatorname{rank}A$.
Over $\mathbb R$ or $\mathbb C$, there are zero, one, or infinitely many solutions. Over a finite field with $q$ elements, a consistent system has $q^{n-r}$ solutions, where $r=\operatorname{rank}A$.
Start with terminology, then Gaussian elimination, free variables, and solution structure.
Parents
Children
- Basic terminologyLinear-system terminology connects linear equations, coefficients, solutions of one equation, and solutions of a...
- Parametric form of the solution set of a system of linear equationsDefinition: The parametric form of the solution set expresses the solutions in terms of one or more free parameters...
- Parametric vector form of the solution set of a system of linear equationsDefinition: The parametric vector form expresses the solution set of a linear system as a vector equation with free...
- Gaussian elimination as a method to solve a linear systemDefinition: Gaussian elimination is a systematic method for solving systems of linear equations by transforming the...
- Definition of echelon form of a linear systemDefinition: The echelon form (or row echelon form ) of a linear system is a form obtained by applying Gaussian...
- All echelon forms of a linear system have the same free variablesTheorem: All echelon forms of a given linear system have the same pivot positions, and therefore the same free...
- Definition of equation operations on a linear systemDefinition: The three equation operations (elementary row operations) on a linear system are: 1. Interchange — Swap...
- Equation operations on a linear system give an equivalent system.Theorem: Applying any of the three equation operations to a linear system produces a new system that is equivalent to...
- Definition of equivalent systems of linear equationsDefinition: Two systems of linear equations are equivalent if they have exactly the same solution set. Equivalently,...
- The geometry of linear systemsThe geometry of linear systems connects algebraic equations with geometric objects. Each linear equation in $n$...
- Definition of consistent linear systemDefinition: A linear system is consistent if it has at least one solution. Equivalently, the system $Ax = b$ is...
- Definition of inconsistent linear systemDefinition: A linear system is inconsistent if it has no solutions. In echelon form of the augmented matrix, an...
- Definition of homogeneous linear system of equationsDefinition: A homogeneous linear system is a system of the form $Ax = 0$, where every constant term on the right-hand...
- Homogeneous linear systems are consistent.Theorem: Every homogeneous linear system $Ax = 0$ is consistent . Proof: The zero vector $x = \mathbf{0}$ always...
- Definition of nontrivial solution to a homogeneous linear system of equationsDefinition: A nontrivial solution to a homogeneous system $Ax = 0$ is any solution $x \neq \mathbf{0}$. In other...
- Definition of trivial solution to a homogeneous linear system of equationsDefinition: The trivial solution to a homogeneous system $Ax = 0$ is the zero vector: $$x = \mathbf{0} = (0, 0,...
- A homogeneous system has a nontrivial solution if and only if it has a free variable.Theorem: A homogeneous system $Ax = 0$ has a nontrivial solution if and only if it has at least one free variable....
- The number of solutions to a linear systemOver $\mathbb R$ or $\mathbb C$, a linear system has zero, one, or infinitely many solutions. An inconsistent system...
- Definition of basic/dependent/leading variable in a linear systemDefinition: In the echelon form of a linear system, a basic variable (also called a dependent or leading variable )...
- Definition of free/independent variable in a linear systemDefinition: In the echelon form of a linear system, a free variable (also called an independent variable ) is a...
- Vector form of solutions to a consistent linear systemTheorem: The general solution to a consistent linear system $Ax = b$ can be expressed in vector form as: $$x = p + c...
- The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.Theorem: The general solution to a nonhomogeneous system $Ax = b$ is: $$x = x p + x h$$ where: - $x p$ is any...
- Definition of ill-conditioned linear systemFor an invertible square system, the condition number $\kappa(A)=\ A\ \ A^{-1}\ $ describes worst-case relative...
No children match.
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.