Linear System of Equation

linear-algebra

Linear System of Equations

Let and be a field. Let , with , and , with .

A system of form

is called a linear system of equations in unknowns .

Homogeneity

Homogeneous

Homogeneous Linear System of Equations

A linear system of equations is called homogeneous if all coefficients are set to zero:

Inhomogeneous

Inhomogeneous Linear System of Equations

A linear system of equations is called inhomogeneous if at least one coefficient is not set to zero:

Matrix Representation

Linear systems of equations can be represented as matrix multiplication:

where is the coefficient matrix, is the unknown variable vector, and is the solution vector.

This means that there exists exactly one solution if is a linear combination of a column of . From this, the Rouché–Capelli theorem follows. follows. follows. follows.

Rouché–Capelli Theorem

Kronecker-Capelli theorem

Definition

Rouché–Capelli Theorem

The Rouché–Capelli theorem is a solvabilty criterion for linear system of equations.

Let and . A linear system of equation is solvable if and only if:

where the matrix is called the extended system matrix of the linear system of equation .

Link to original

Solving

To describe all solutions of a linear system of equations, we use the interpretation of matrix multiplication as a linear mapping:

We assume that the linear system of equations has a solution . If is any other solution, it follows from that: