Appendix:Glossary of linear algebra

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Glossary of linear algebra

Wikipedia

This is a glossary of linear algebra.

affine transformation

A linear transformation between vector spaces followed by a translation.

basis

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

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.

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.

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.

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.

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.

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 ${\displaystyle \mathbf {v} ,1^{*}(\mathbf {v} )=\mathbf {v} }$.