# Linear Algebra/Glossary

## A

Also called classical adjoint. The matrix adj A formed from a square matrix A by replacing the (i, j)-entry of A by the (i, j)-cofactor, for all i and j, and then transposing the resulting matrix.
affine transformation
A mapping ${\displaystyle T:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}}$  of the form ${\displaystyle T(x)=Ax+b}$ , with A an ${\displaystyle m\times n}$  matrix and b in ${\displaystyle \mathbb {R} ^{m}}$ .
algebraic multiplicity
The multiplicity of an eigenvalue as a root of the characteristic equation.
angle
Between nonzero vectors u and v in ${\displaystyle \mathbb {R} ^{2}}$  and ${\displaystyle \mathbb {R} ^{3}}$ . The angle ${\displaystyle \vartheta }$  between the two directed line segments from the origin to the points u and v. Related to the scalar product by ${\displaystyle u\bullet v=\left\Vert u\right\Vert \left\Vert v\right\Vert \cos \vartheta }$

## C

coefficients
A constant by which a variable are multiplied, fx. ${\displaystyle 2\,}$  are a coefficient in the following equation ${\displaystyle 2x=4\,}$ .
consistent linear system
A linear system with at least one solution.

## E

equivalent linear systems
Linear systems with the same solution set.
echelon form
Also called row echelon form. An echelon matrix that is row equaivalent to the given matrix.
echelon matrix
Also called row echelon matrix. A rectangular matrix that has three properties: (1) All nonzero rows are above any row of all zeros. (2) Each leading entry of a row is in a column to the right of the leading entry of the row above it. (3) All entries in a column below a leading entry are zero.

## I

inconsistent linear system
A linear system with no solution.

## L

linear equation
An equation that can be written in the form ${\displaystyle a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=b\,}$ , where b and the coefficients ${\displaystyle a_{1},...,a_{n}\,}$  are real or complex numbers.
linear system
A collection of one or more linear equations involving the same variables, fx. ${\displaystyle x_{1},...,x_{n}\,}$ .

## R

reduced echelon form
Also called reduced row echelon form. A reduced echelon matrix that is row equivalent to a given matrix.
reduced echelon matrix
A rectangular matrix in echelon form that has these additional properties: The leading entry in each nonzero row is 1, and each leading 1 is the only nonzero entry in its column.
reduced row echelon form
See reduced echelon form.
row echelon form
See echelon form.
row echelon matrix
See echelon matrix.

## S

solution
A list ${\displaystyle \left(s_{1},s_{2},...,s_{n}\right)\,}$  of numbers that makes each equation in system a true statement when the values ${\displaystyle s_{1},...,s_{n}\,}$  are substituted for ${\displaystyle x_{1},...,x_{n}\,}$  respectively.
solution set
The set of all possible solutions of a linear system. The solution set is empty when the linear system is inconsistent.
system of linear equations
Also called a linear system, is a collection of one or more linear equations involving the same set of variables, fx. ${\displaystyle x_{1},...,x_{n}\,}$ .