Linear System of Equation

linear-algebra

Linear System of Equations

Let $m,n\in {\mathbb{N}}_{\ge 1}$ and $F$ be a field. Let $a_{ij}\in F$, with $1\le i\le m,1\le j\le n$, and $b_i\in F$, with $1\le i\le m$.

A system of form

$$\begin{array}{rl}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n& =b_1\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n& =b_2\\ & ⋮\\ a_{m1}{x}_1+a_{m2}{x}_2+\cdots +a_{mn}{x}_n& =b_m\end{array}$$

is called a linear system of equations in unknowns $x_1,x_2,\dots ,x_n$.

Homogeneity

Homogeneous

Homogeneous Linear System of Equations

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

$${\mathbf{b}}_1,{\mathbf{b}}_2,\dots ,{\mathbf{b}}_n=0$$

Inhomogeneous

Inhomogeneous Linear System of Equations

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

$$\exists b_i:b_i\ne 0\forall i\in \{1,\dots ,n\}$$

Matrix Representation

Linear systems of equations can be represented as matrix multiplication:

$$A\cdot \mathbf{x}=\mathbf{b}$$

where $A\in F^{m\times n}$ is the coefficient matrix, $\mathbf{x}\in F^n$ is the unknown variable vector, and $\mathbf{b}\in F^n$ is the solution vector.

This means that there exists exactly one solution $\mathbf{x}$ if $\mathbf{b}$ is a linear combination of a column of $A$. 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 $A\in F^{m\times n}$ and $\mathbf{b}\in F^n$. A linear system of equation $A\cdot \mathbf{x}=\mathbf{b}$ is solvable if and only if:

$$\mathrm{rank}(A)=\mathrm{rank}(A\cdot \mathbf{b})$$

where the matrix $(A\cdot \mathbf{b})$ is called the extended system matrix of the linear system of equation $A\cdot \mathbf{x}=\mathbf{b}$.

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:

$$f(\mathbf{x})=A\cdot \mathbf{x}$$

We assume that the linear system of equations $f(\mathbf{x})=A\cdot \mathbf{x}=\mathbf{b}$ has a solution ${\mathbf{x}}_0$. If $\mathbf{x}$ is any other solution, it follows from $f({\mathbf{x}}_0)=f(\mathbf{x})=\mathbf{b}$ that:

$$f(\mathbf{x}-{\mathbf{x}}_0)=A\cdot (\mathbf{x}-{\mathbf{x}}_0)=0$$