Glossary of Linear Algebra Terms

## Glossary of Linear Algebra Terms

basis for a subspace:
A basis for a subspace W is a set of vectors v1, ...,vk in W such that:

1. v1, ..., vk are linearly independent; and
2. v1, ..., vk span W.

characteristic polynomial of a matrix:
The characteristic polynomial of a n by n matrix A is the polynomial in t given by the formula det(A - t*I).

column space of a matrix:
The column space of a matrix is the subspace spanned by the columns of the matrix considered as vectors. See also: row space.

consistent linear system:
A system of linear equations is consistent if it has at least one solution. See also: inconsistent.

defective matrix:
A matrix A is defective if A has an eigenvalue whose geometric multiplicity is less than its algebraic multiplicity.

diagonalizable matrix:
A matrix is diagonalizable if it is dimension of a subspace:
The dimension of a subspace W is the number of vectors in any basis of W. (If W is the subspace {0}, we say that its dimension is 0.)

row echelon form of a matrix:
A matrix is in row echelon form if:

1. all rows that consist entirely of zeros are grouped together at the bottom of the matrix; and
2. the first (counting left to right) nonzero entry in each nonzero row appears in a column to the right of the first nonzero entry in the preceding row (if there is a preceding row).

reduced row echelon form of a matrix:
A matrix is in reduced row echelon form if:

1. the matrix is in row echelon form;
2. the first nonzero entry in each nonzero row is the number 1; and
3. the first nonzero entry in each nonzero row is the only nonzero entry in its column.

eigenspace of a matrix:
The eigenspace associated with the eigenvalue c of a matrix A is the null space of A - c*I.

eigenvalue of a matrix:
An eigenvalue of a n by n matrix A is a scalar c such that A*x = c*x holds for some nonzero vector x (where x is an n-tuple). See also: eigenvector.

eigenvector of a matrix:
An eigenvector of a n by n matrix A is a nonzero vector x such that A*x = c*x holds for some scalar c. See also: eigenvalue.

equivalent linear systems:
Two systems of linear equations in n unknowns are equivalent if they have the same set of solutions.

homogeneous linear system:
A system of linear equations A*x = b is homogeneous if b = 0.

inconsistent linear system:
A system of linear equations is inconsistent if it has no solutions. See also: consistent.

inverse of a matrix:
The matrix B is an inverse for the matrix A if A*B = B*A = I.

invertible matrix:
A matrix is invertible if it has an inverse.

least-squares solution of a linear system:
A least-squares solution to a system of linear equations A*x = b is a vector x that minimizes the length of the vector A*x - b.

linear combination of vectors:
A vector v is a linear combination of the vectors v1, ..., vk if there exist scalars a1, ..., ak such that v = a1*v1 + ... + ak*vk.

linearly dependent vectors:
The vectors v1, ..., vk are linearly dependent if the equation a1*v1 + ... + ak*vk = 0 has a solution where not all the scalars a1, ..., ak are zero.

linearly independent vectors:
The vectors v1, ..., vk are linearly independent if the only solution to the equation a1*v1 + ... + ak*vk = 0 is the solution where all the scalars a1, ..., ak are zero.

linear transformation :
A linear transformation from V to W is a function T from V to W such that:

1. T(u+v) = T(u) + T(v) for all vectors u and v in V; and
2. T(a*v) = a*T(v) for all vectors v in V and all scalars a.

algebraic multiplicity of an eigenvalue:
The algebraic multiplicity of an eigenvalue c of a matrix A is the number of times the factor (t-c) occurs in the characteristic polynomial of A.

geometric multiplicity of an eigenvalue:
The geometric multiplicity of an eigenvalue c of a matrix A is the dimension of the eigenspace of c.

nonsingular matrix:
An n by n matrix A is nonsingular if the only solution to the equation A*x = 0 (where x is an n-tuple) is x = 0. See also: singular.

null space of a matrix:
The null space of a m by n matrix A is the set of all n-tuples x such that A*x = 0.

null space of a linear transformation:
The null space of a linear transformation T is the set of vectors v in its domain such that T(v) = 0.

nullity of a matrix:
The nullity of a matrix is the dimension of its null space.

nullity of a linear transformation:
The nullity of a dimension of its null space.

orthogonal set of vectors:
A set of n-tuples is orthogonal if the dot product of any two of them is 0.

orthogonal matrix:
A matrix A is orthogonal if A is invertible and its orthogonal linear transformation:
A linear transformation T from V to W is orthogonal if T(v) has the same length as v for all vectors v in V.

orthonormal set of vectors:
A set of n-tuples is orthonormal if it is orthogonal and each vector has length 1.

range of a matrix:
The range of a m by n matrix A is the set of all m-tuples A*x, where x is any n-tuple.

range of a linear transformation:
The range of a linear transformation T is the set of all vectors T(v), where v is any vector in its domain.

rank of a matrix:
The rank of a matrix is the number of nonzero rows in any row equivalent matrix that is in row echelon form.

rank of a linear transformation:
The rank of a linear transformation (and hence of any matrix regarded as a linear transformation) is the dimension of its range. Note: A theorem tells us that the two definitions of rank of a matrix are equivalent.

row equivalent matrices:
Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations:
The elementary row operations performed on a matrix are:

• interchange two rows;
• multiply a row by a nonzero scalar;
• add a constant multiple of one row to another.

row space of a matrix:
The row space of a matrix is the subspace spanned by the rows of the matrix considered as vectors. See also: similar matrices:
Matrices A and B are similar if there is a square nonsingular matrix S such that S^(-1)*A*S = B.

singular matrix:
An n by n matrix A is singular if the equation A*x = 0 (where x is an n-tuple) has a nonzero solution for x. See also: nonsingular.

span of a set of vectors:
The span of the vectors v1, ..., vk is the subspace V consisting of all linear combinations of v1, ..., vk. One also says that the subspace V is spanned by the vectors v1, ..., vk and that these vectors span V.

subspace:
A subset W of n-space is a subspace if:

1. the zero vector is in W;
2. x+y is in W whenever x and y are in W; and
3. a*x is in W whenever x is in W and a is any scalar.

symmetric matrix:
A matrix is symmetric if it equals its transpose.