linear system

A linear system is a collection of two or more linear equations involving the same variables.

A solution is an assignment of values to the variables such that all equations are simultaneously satisfied.

A finite linear system of $m$ linear equations and $n$ unknowns and coefficients can be written

{\begin{cases} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} = b_{1} \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} = b_{2} \\ ⋮ \\ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} = b_{m} \end{cases}

where $x_{i}$ are the unknowns, $a_{i j}$ are the coefficients of the system, and $b_{i}$ are the constant terms.

Equivalently, the vector equation can be constructed

x_{1} (\begin{matrix} a_{11} \\ a_{21} \\ ⋮ \\ a_{m 1} \end{matrix}) + x_{2} (\begin{matrix} a_{12} \\ a_{22} \\ ⋮ \\ a_{m 2} \end{matrix}) + \dots + x_{n} (\begin{matrix} a_{1 n} \\ a_{2 n} \\ ⋮ \\ a_{m n} \end{matrix}) = (\begin{matrix} b_{1} \\ b_{2} \\ ⋮ \\ b_{m} \end{matrix})

which expresses the unknowns as weights for column vectors in a linear combination.

The linear system can also be written in the form

(\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}) (\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}) = (\begin{matrix} b_{1} \\ b_{2} \\ ⋮ \\ b_{m} \end{matrix})

known as the matrix equation, commonly written $A \vec{x} = \vec{b}$ , where $\vec{x}$ is a column vector containing $n$ unknowns, $\vec{b}$ is a column vector containing $m$ constants, and $A$ is a matrix of dimension $m \times n$ containing the coefficients.

Often, this matrix equation is written as an augmented matrix, for the purposes of performing Gaussian elimination.

A solution of a linear system is the assignment of values to the variables such that each of the equations is satisfied.

The set of all possible solutions is called the solution set;
as vectors, this set is an $m$ -dimensional vector space.

For any linear system of equations, there are 3 possibilities:

the system has exactly 1 solution
the system has infinitely many solutions
the system is inconsistent, i.e. there are no solutions