Linear Algebra/Glossary

A

 * adjugate : Also called classical adjoint. The matrix adj A formed from a square matrix A by replacing the (i, j)-entry of A by the (i, j)-cofactor, for all i and j, and then transposing the resulting matrix.
 * affine transformation : A mapping $$T:\mathbb{R}^n\rightarrow\mathbb{R}^m$$ of the form $$T(x)=Ax+b$$, with A an $$m \times n$$ matrix and b in $$\mathbb{R}^m$$.
 * algebraic multiplicity : The multiplicity of an eigenvalue as a root of the characteristic equation.
 * angle : Between nonzero vectors u and v in $$\mathbb{R}^2$$ and $$\mathbb{R}^3$$. The angle $$\vartheta$$ between the two directed line segments from the origin to the points u and v. Related to the scalar product by $$u \bullet v=\left \Vert u \right \Vert \left \Vert v \right \Vert \cos \vartheta$$

C

 * coefficients : A constant by which a variable are multiplied, fx. $$2\,$$ are a coefficient in the following equation $$2x=4\,$$.
 * consistent linear system : A linear system with at least one solution.

E

 * equivalent linear systems : Linear systems with the same solution set.
 * echelon form : Also called row echelon form. An echelon matrix that is row equaivalent to the given matrix.
 * echelon matrix : Also called row echelon matrix. A rectangular matrix that has three properties: (1) All nonzero rows are above any row of all zeros. (2) Each leading entry of a row is in a column to the right of the leading entry of the row above it. (3) All entries in a column below a leading entry are zero.

I

 * inconsistent linear system : A linear system with no solution.

L

 * linear equation : An equation that can be written in the form $$a_1x_1+a_2x_2+...+a_nx_n=b\,$$, where b and the coefficients $$a_1,...,a_n\,$$ are real or complex numbers.
 * linear system : A collection of one or more linear equations involving the same variables, fx. $$x_1,...,x_n\,$$.

R

 * reduced echelon form : Also called reduced row echelon form. A reduced echelon matrix that is row equivalent to a given matrix.
 * reduced echelon matrix : A rectangular matrix in echelon form that has these additional properties: The leading entry in each nonzero row is 1, and each leading 1 is the only nonzero entry in its column.
 * reduced row echelon form : See reduced echelon form.
 * row echelon form : See echelon form.
 * row echelon matrix : See echelon matrix.

S

 * solution : A list $$\left(s_1,s_2,...,s_n\right)\,$$ of numbers that makes each equation in system a true statement when the values $$s_1,...,s_n\,$$ are substituted for $$x_1,...,x_n\,$$ respectively.
 * solution set : The set of all possible solutions of a linear system. The solution set is empty when the linear system is inconsistent.
 * system of linear equations : Also called a linear system, is a collection of one or more linear equations involving the same set of variables, fx. $$x_1,...,x_n\,$$.