# Appendix:Glossary of linear algebra

This is a glossary of linear algebra.

Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

## A

affine transformation
A linear transformation between vector spaces followed by a translation.

## B

basis
In a vector space, a linearly independent set of vectors spanning the whole vector space.

## D

determinant
The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of ${\displaystyle 1}$ for the unit matrix.
diagonal matrix
A matrix in which only the entries on the main diagonal are non-zero.
dimension
The number of elements of any basis of a vector space.

## I

identity matrix
A diagonal matrix all of the diagonal elements of which are equal to ${\displaystyle 1}$.
inverse matrix
Of a matrix ${\displaystyle A}$, another matrix ${\displaystyle B}$ such that ${\displaystyle A}$ multiplied by ${\displaystyle B}$ and ${\displaystyle B}$ multiplied by ${\displaystyle A}$ both equal the identity matrix.

## L

linear algebra
The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems of linear equations.
linear combination
A sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element).
linear equation
A polynomial equation of the first degree (such as ${\displaystyle x=2y-7}$).
linear transformation
A map between vector spaces which respects addition and multiplication.
linearly independent
(Of a set of vectors or ring elements) whose nontrivial linear combinations are nonzero.

## M

matrix
A rectangular arrangement of numbers or terms having various uses such as transforming coordinates in geometry, solving systems of linear equations in linear algebra and representing graphs in graph theory.

## S

spectrum
Of a bounded linear operator ${\displaystyle A}$, the scalar values ${\displaystyle \lambda }$ such that the operator ${\displaystyle A-\lambda I}$, where ${\displaystyle I}$ denotes the identity operator, does not have a bounded inverse.
square matrix
A matrix having the same number of rows as columns.

## V

vector
A directed quantity, one with both magnitude and direction; an element of a vector space.
vector space
A set ${\displaystyle V}$, whose elements are called "vectors", together with a binary operation ${\displaystyle +}$ forming a module ${\displaystyle (V,+)}$, and a set ${\displaystyle F^{*}}$ of bilinear unary functions ${\displaystyle f^{*}:V\rightarrow V}$, each of which corresponds to a "scalar" element ${\displaystyle f}$ of a field ${\displaystyle F}$, such that the composition of elements of ${\displaystyle F^{*}}$ corresponds isomorphically to multiplication of elements of ${\displaystyle F}$, and such that for any vector ${\displaystyle \mathbf {v} ,1^{*}(\mathbf {v} )=\mathbf {v} }$.