Linear algebra

number that can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit

branch of mathematics that studies vector spaces and linear transformations

geometric object that has magnitude (or length) and direction

MATLAB (Matrix Laboratory) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.

part of a line that is bounded by two distinct end points; line with two endpoints

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse.

collection of linear equations involving the same set of variables

generalization of Euclidean geometry to higher-dimensional vector spaces

inner product space that is metrically complete; a Banach space whose norm induces an inner product (follows the parallelogram identity)

single field element, e.g. real number, in the context of linear algebra

vectors that map to their scalar multiples, and the associated scalars

subset of a vector space that allows defining coordinates

vector of length one

a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics

inequality of the form |x + y| ≤ |x| + |y|

property of a set of vectors of a vector space

length in a vector space

notation for quantum states

measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by a matrix

sum of the elements of the main diagonal of a square matrix

expression constructed from a set of terms by the addition of terms in the set to each other any number of times

thumb|200px|right|class=skin-invert-image|The transpose AT of a matrix A can be obtained by reflecting the elements along its main diagonal. Repeating the process on the transposed matrix returns the elements to their original position.

square matrix with non-zero determinant

method for orthonormalising a set of vectors

computer algebra system

mnemonic rule for evaluating the determinant of 3x3 matrices

number of vectors in any basis of the vector space

homogeneous polynomial of degree two in a number of variables

complex matrix A* obtained from a matrix A by transposing it and conjugating each entry

thumb|Two intersecting planes: Two-dimensional planes are the hyperplanes in three-dimensional space.

vector space of linear functionals (may consist only on continuous functionals or of all functionals)

inverse image of zero under a linear map

smallest vector subspace containing a given subspace

linear functional on tensor product square of a vector space

geometric structure that generalizes the Euclidean space

type of matrix

subset of a vector space that forms a vector space itself

polynomial with roots that are the eigenvalues of a given matrix

antisymmetric permutation object acting on tensors

function with multiplicative scaling behaviour

notions of sums for matrices in linear algebra

for a square matrix, the transpose of the cofactor matrix

coordinate system in which the location of a point of a simplex is specified as the center of mass, or barycenter, of usually unequal masses placed at its vertices

matrix decomposition

Mathcad is computer software for the verification, validation, documentation and re-use of mathematical calculations in engineering and science, notably mechanical, chemical, electrical, and civil engineering. Released in 1986 for MS-DOS, it introduced live editing (WYSIWYG) of typeset mathematical notation in an interactive notebook, combined with automatic computations. It was originally developed by Mathsoft, and since 2006 has been a product of Parametric Technology Corporation.

function valued in a vector space; typically a real or complex one

theorem on extension of bounded linear functionals

set of functions of a given kind from a set X to a set Y; it is called a space because in many applications it is a topological space (including metric spaces), a vector space, or both

norm on a vector space of matrices

n x n invertible matrices over a ring

mathematics

linear mapping from a vector space into its field of scalars

linear transformation that, when applied multiple times to any value, gives the same result as if it were applied once

thumb|right|A loop of wire (black), carrying a electric current|current I, creates a [[magnetic field B (blue). If the position and current of the wire are reflected across the plane indicated by the dashed line, the magnetic field it generates would not be reflected: Instead, it would be reflected and reversed. The position and current at any point in the wire are "true" vectors, but the magnetic field B is a pseudovector.]]

change of coordinates for a vector space

basis of a normed space consisting of mutually orthogonal elements of norm 1

form of a matrix indicating its eigenvalues and their algebraic multiplicities

matrix which, when multiplied by itself, yields itself

methods in computer graphics to project three-dimensional objects onto a plane by means of numerical calculations

element of the real commutative associative algebra ℝ[j] / (j² − 1), i.e. the reals with an extra square root of +1 adjoined