A linear combination of $x_1,x_2,\dots ,x_n$ is an expression

where $x_i$'s are indeterminates or variables; and $a_i$'s are coefficients and belong to a field $𝕂$

A linear equation in the set of variables $𝒳$, where without loss of generality $𝒳=\{x_1,x_2,\dots ,x_n\}$ is an expression

The value $b_i$ is the constant of the linear equation and belongs to the field $𝕂$ that contains the constants.

A linear equation

is called homogeneous if $b=0$.

An $n$-tuple $(s_1,s_2,\dots ,s_n)\in {𝕂}^n$ is a solution (order is important) to the linear equation

if and only if

If the set $𝒮$ of solution to a linear equation

is not empty then the equation is consistent, else it is inconsistent

A system of linear equation in the set of variables $𝒳=\{x_1,x_2,\dots ,x_n\}$ is a set of linear equations

An $n$-tuple $(s_1,s_2,\dots ,s_n)\in {𝕂}^n$ is a solution to a system of linear equations if it is solution to each equation in the system of linear equations.

An system of linear equations is consistent if its solution set is non-empty; otherwise it is inconsistent.

An system of linear equations is homogeneous if each of its equations is homogeneous.

Two system of linear equations are equivalent if they have the same set of solutions.

An $m\times n$ matrix matrix $A={\left\{a_{ij}\right\}}_{1\le i\le m,1\le j\le n}$ is a rectangular array of numbers with $m$ rows and $n$ columns. Each $a_{ij}$ is called entry.

Let $A={\left\{a_{ij}\right\}}_{1\le i\le m,1\le j\le n}$ and $B={\left\{b_{ij}\right\}}_{1\le i\le r,1\le j\le p}$ be two matrices. They are equal

if $m=r$, $n=p$ and for all $1\le i\le m,1\le j\le n$ we have

In each row of a system of linear equation, the first variable with a non-zero coefficient is called leading variable.

A system is in Echelon form if each leading variable in Echelon form if each leading variable is to the right of the leading variable in the row above it, except for the leading variable in the first row, and any all-zero rows are at the bottom.

The non-leading variables in Echelon form are called free variables.

Let $A=\left\{a_{ij}\right\}$ be an $m\times n$ matrix and $\beta$ be a scalar. Define

Let $A=\left\{a_{ij}\right\}$ and $B=\left\{b_{ij}\right\}$ be two $m\times n$ matrices. Define

Let $A=\left\{a_{ij}\right\}$ be an $m\times n$ matrix and $B=\left\{b_{ij}\right\}$ be an $n\times k$ matrix. Define

where

A linear combination of vectors ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,\dots ,{\overrightarrow{v}}_n$ is an expression

where ${\alpha }_i$'s are coefficients and belong to $𝕂$

Let $C=AB$ then

• rows of $C$ are linear combinations of the rows of $B$
• columns of $C$ are linear combinations of the columns of $A$

Let $A=\left\{a_{ij}\right\}$ be an $m\times n$ matrix. The transpose of $A$ denoted by $A^T$ is an $n\times m$ matrix $\left\{a_{ij}^T\right\}$ where for all $1\le i\le \mathrm{n}$ and for all $1\le j\le \mathrm{m}$

Let $A$ be a non-zero square matrix. If there is non-zero square matrix $B$ such that

then $A$ is called zero divisor.

A square matrix $A$ is invertible if there is a matrix $B$ such that

Then $B$ is called the inverse of $A$ and denoted by $A^{-1}$.

The elementary row operations, (also row operations, Gaussian operations) are

• row swapping
• rescaling (multiplication with a non-zero constant)
• row combinations (adding a multiple of another row)