Appendix:Glossary of linear algebra

This is a glossary of linear algebra.

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 $1$ 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 $1$.

inverse matrix

Of a matrix $A$, another matrix $B$ such that $A$ multiplied by $B$ and $B$ multiplied by $A$ 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 $x=2y-7$).

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 $A$, the scalar values $\lambda$ such that the operator $A-\lambda I$, where $I$ 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 $V$, whose elements are called "vectors", together with a binary operation $+$ forming a module $(V,+)$, and a set $F^{*}$ of bilinear unary functions $f^{*}:V\rightarrow V$, each of which corresponds to a "scalar" element $f$ of a field $F$, such that the composition of elements of $F^{*}$ corresponds isomorphically to multiplication of elements of $F$, and such that for any vector $\mathbf {v} ,1^{*}(\mathbf {v} )=\mathbf {v}$.