adjoint
The transpose of a square matrix obtained by replacing each element with its cofactor. Each element aij is replaced with the cofactor of the element in aji. Also called adjugate.
augmented matrix
A matrix consisting of the coefficients of a system of linear equations and a final column containing the constants. Each column contains coefficients of the same variable from the system.
basis
A set of vectors, {v1,v2,v3,...}, which spans a vector space, V, and has the property that the vectors are linearly independent.
characteristic equation
The linear equation obtained by setting the determinant of the matrix formed by A -λI equal to 0. The symbol λ represents the eigenvalues of the matrix A.
characteristic polynomial
The determinant of the square matrix defined by subtracting a multiple of an identity matrix from a transformation matrix, A -λI.
closure
The property of number systems and vector spaces in which particular operations used in the system produce only elements of that system or vector space. The whole numbers are closed to addition, but the whole numbers are not closed to division.
coefficient matrix
A matrix consisting of all the coefficients of the variables in a system of linear equations.
cofactors of a matrix
Created from a matrix by multiplying the determinants of the minors of the matrix by powers of -1. The cofactor Aij of matrix A is equal to (-1)^(i+j) det(Mij), where Mij is a minor of the matrix.
column matrix (vector)
An m x 1 matrix; a matrix with one column and m rows.
commutativity
The property of addition and multiplication that preserves the final result when the elements in the operation are reversed in order. Multiplying 4 x 5 yields the same result as multiplying 5 x 4.
consistent system of equations
A system of equations that has at least one solution.
contraction
A function or transformation in which distances are shortened. The function f performed on the element u is a contraction if f(u)=ru, where 0<r<1.
coordinate vector
A column or row vector of real numbers a1, a2,...ak, which are used to express the ordered basis S=v1, v2,...,vk in the form a1v1, a2v2,...,akvk.
dependent system of equations
A system of equations in which at least one of the equations is a linear combination of one or more of the other equations in the system.
determinant
The sum of all the products of all the possible permutations of the elements of a square matrix. Each product is multiplied by either +1 or -1.
diagonal matrix
A square matrix in which all the elements not on the main diagonal are equal to zero.
diagonalizable matrix
A matrix for which you can construct another matrix similar to that original matrix that is a diagonal matrix.
dilation
A function or transformation in which distances are lengthened. The function f performed on the element u is a dilation if f(u)=ru, where r>1.
dimension of matrix
The number of rows and columns of a matrix, expressed in the format m x n. The dimension 2 x 3 indicates a matrix with two rows and three columns.
dimension of a vector space
The number of vectors in the basis of the vector space.
dot product (inner product) of two vectors
The sum of the products of the corresponding elements of the vectors.
eigenvalue
A number associated with a matrix in which multiplying a vector by the matrix is the same as multiplying that vector by the eigenvalue; the resulting vector is a multiple of the original vector multiplied by the eigenvalue. Given the square matrix A, vector x, and eigenvalue λ, then Ax=λx.
eigenvector
A vector x associated with a square matrix A such that Ax=λx, where λ is some scalar. The scalar λ is called an eigenvalue of A; you say that x is an eigenvector corresponding of λ.
equivalent matrices
Two m x n matrices in which one is obtained from the other by performing elementary row operations.
free variable
A variable in a system of equations for which the other variables can be written in terms of that variable. The variable x3 is a free variable if x1=2x3 and x2=3x3.
geometric mean
The number between two other numbers that is the positive square root of the two numbers. The geometric mean of 4 and 9 is 6, because 6 is the square root of 4 x 9=36.
homogeneous system of equations
A system of equations, all set equal to zero. The system always has a solution; the solution is trivial if all the variables are equal to zero and nontrivial if some of the variables are not equal to zero.
identity
An element in a set associated with an operation such that performing the operation with another element and the identity does not change the value or state of that other element. In the real number system, the additive identity is 0 and the multiplicative identity is 1.
identity matrix
A square matrix with its main diagonal (from upper left to lower right) consisting of 1s and the rest of the elements 0s.
image
The resulting matrix or vector after performing a function operation on a matrix or vector.
inconsistent system of equations
A linear system that has no solution.
independent equations
A system of equations for which none of the equations is a linear combination of the other equations in the system.
index of element in matrix
The row and column designation or position of the element a, indicated by a subscript aij. The element a24 is in the second row and fourth column of the matrix.
inverse matrix
A square matrix associated with another square matrix such that the product of the matrix and its inverse is an identity matrix.
inversion (of a permutation)
A permutation in which a larger integer precedes a smaller integer. When considering the permutations of the first three positive integers, the permutation 132 has one inversion (3 comes before 2), and the permutation 321 has three inversions (3 comes before 2, 3 comes before 1, 2 comes before 1).
invertible matrix
A square matrix that has an inverse. Also called a nonsingular matrix.
kernel of linear transformation
The subset of a vector space for which a linear transformation takes all the vectors to a 0 vector.
linear combination
The sum of the products of two or more quantities, each multiplied by some constant value. The linear combination of the elements of a set of vectors and some scalars is written a1v1+a2v2+...akvk, where each ai represents a real number and each vi represents a vector.
linear equation
An equation in the form a1x1+a2x2+...+akxk=b, where the as are constants, the xs are unknowns, and the b is a constant.
linear independence
When the linear combination of a set of elements is equal to 0 only when each of the scalar multiples is 0. The set of vectors {v1, v2, ...vk} is linearly independent when a1v1+a2v2+...+akvk=0 only if a1=a2=...=ak=0.
linear transformation (operator)
A transformation or operation performed on elements from a set in which addition and scalar multiplication are preserved. Letting T represent the transformation, u and v represent vectors, and a and b represent scalars, then T(au+bv)=aT(u)+bT(v).
magnitude of vector
The length of a vector. The value obtained by computing the square root of the sum of the squares of the elements of the vector.
main diagonal of matrix
The elements of a square matrix in which the two elements of the index are the same: a11, a22,...,akk. These elements start in the upper left-hand corner and run down to the lower right-hand corner of the matrix.
minor of matrix
A subset of a matrix delineated by removing a row and column associated with a particular element. A sub-matrix is related to the aij element of the original matrix in that the ith row and jth column are eliminated.
natural basis of vectors
The basis of a set of vectors where one element in each vector is equal to 1 and the other elements are equal to 0, and no vector has the 1 in the same position. See also standard basis of matrices.
noninvertible matrix
A square matrix that does not have an inverse. Also called a singular matrix.
nonsingular matrix
A square matrix that has an inverse. Also called an invertible matrix.
non-trivial solution
A solution of a homogeneous system of equations in which the values of the scalars are not all zero.
null space
The set of all the solutions of a system of homogeneous equations.
orthogonal
Vectors whose inner product is 0. An orthogonal basis of a vector space is one in which all the vectors are orthogonal to one another.
orthonormal
An orthogonal basis of a vector set in which the vectors all have a magnitude of 1.
parametric equation
An equation in which a parameter is used to distinguish between different cases. In the slope-intercept form of the linear equation, y=mx+b, the m and b are parameters, affecting the slant and y-intercept of the graph of the line.
permutation
A rearrangement of an ordered listing of elements. The six permutations of the first three letters of the alphabet are abc, acb, bac, bca, cab, and cba.
range of linear transformation
All the vectors that are results of performing a linear transformation on a vector space.
reciprocal
Any real number (except 0) raised to the -1 power. The reciprocal of 2 is 1/2. The reciprocal of 4/3 is 3/4. The product of a number and its reciprocal is 1.
reverse diagonal
The diagonal of a square matrix that runs from the upper right corner to the lower-left corner of the matrix.
semiperimeter
Half the perimeter of a polygon.
singular matrix
A square matrix that does not have an inverse. Also called a noninvertible matrix.
skew-symmetric matrix
A matrix in which each element aij=-aij. The matrix is equal to the negative of its transpose. If the element in the second row, third column is 4, then the element in the third row, second column is -4.
span
All the vectors in a vector space, V, that are linear combinations of a set of vectors, {v1, v2, ..., vk}, called its spanning set.
spanning set
A set of vectors, {v1, v2,...vk}, whose linear combinations produce the vectors in a vector space.
standard basis of matrices
A basis consisting of vectors that are columns of an identity matrix. Each vector has one nonzero element, which is a 1.
standard position of a vector
A vector positioned with its endpoint at the origin.
subspace
A non-empty vector space that is a subset of a vector space with the same operations as the vector space.
transition matrix
A matrix used in the rule or process defined by a transformation.
transpose of matrix
A matrix constructed by changing all rows of a matrix to columns and all columns to rows. Each element aij becomes element aji.
triangular matrix
A matrix in which all the elements either above or below the main diagonal are 0. An upper triangular matrix has all zeros below the main diagonal, and a lower triangular matrix has all zeros above the main diagonal.
trivial solution
When the solution of a homogeneous system has each variable equal to 0.
unit vector
A vector whose magnitude is equal to 1.
vector space
A set of elements subject to properties involving two defined operations, + and x, and including identities and inverses.
zero matrix
An m x n matrix in which each element is 0.