Appendix:Glossary of linear algebra

This is a glossary of linear algebra.

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