MATH 310: Applied Linear Algebra

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Glossary of Linear Algebra Terms

Thanks to Gene Herman for compiling this Glossary as part of his Math 215 Homepage at Grinnell University.

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.

basis for a subspace:

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

  1. {v1, ..., vk} is linearly independent; and
  2. {v1, ..., vk} spans 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 - tI).

column space of a matrix:

The column space of a matrix is the subspace spanned by the columns of the matrix considered as a set of 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 similar to a diagonal matrix.
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.)

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

An eigenvalue of a square matrix A is a scalar c such that Ax = cx holds for some nonzero vector x. See also: eigenvector.

eigenvector of a matrix:

An eigenvector of a square matrix A is a nonzero vector x such that Ax = cx holds for some scalar c. See also: eigenvalue.

elementary matrix:
An elementary matrix is a matrix that is obtained by performing an elementary row operation on an identity matrix.
equivalent linear systems:
Two systems of linear equations in n unknowns are equivalent if they have the same set of solutions.
geometric multiplicity of an eigenvalue:

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

homogeneous linear system:
A system of linear equations Ax = 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 AB = BA = 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 Ax = b is a vector x that minimizes the length of the vector Ax - 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 = a1v1+ ...+ akvk.

linear dependence relation for a set of vectors:

A linear dependence relation for the set of vectors {v1, ..., vk} is an equation of the form a1v1+ ...+ akvk = 0, where not all the scalars a1, ..., ak are zero.

linearly dependent set of vectors:
The set of vectors {v1, ..., vk} is linearly dependent if the equation a1v1+ ...+ akvk = 0 has a solution where not all the scalars a1, ..., ak are zero (i.e., if {v1, ..., vk} satisfies a linear dependence relation).
linearly independent set of vectors:

The set of vectors {v1, ..., vk} is linearly independent if the only solution to the equation a1v1+ ...+ akvk = 0 is the solution where all the scalars a1, ..., ak are zero. (i.e., if {v1, ..., vk} does not satisfy any linear dependence relation).

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(av) = aT(v) for all vectors v in V and all scalars a.
nonsingular matrix:

A square matrix A is nonsingular if the only solution to the equation Ax = 0 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 vectors x in Rn such that Ax = 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 linear transformation is the dimension of its null space.
orthogonal complement of a subspace:
The orthogonal complement of a subspace S of Rn is the set of all vectors v in Rn such that v is orthogonal to every vector in S.
orthogonal set of vectors:

A set of vectors in Rn 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 inverse equals its transpose; i.e., A-1 = AT.
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 vectors in Rn is orthonormal if it is an orthogonal set and each vector has length 1.
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 A is the number of nonzero rows in the reduced row echelon form of A;
i.e., the dimension of the row space of A.
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.
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.
row equivalent matrices:
Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations.
row operations:

The elementary row operations which can be 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 a set of vectors. See also: column space.

similar matrices:

Matrices A and B are similar if there is a square invertible matrix S such that S-1AS = B.

singular matrix:

A square matrix A is singular if the equation Ax = 0 has a nonzero solution for x. See also: nonsingular.

span of a set of vectors:

The span of the set of 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 set of vectors {v1, ..., vk} and that this set of vectors spans V.

subspace:

A subset W of Rn is a subspace of Rn if:

  1. the zero vector is in W;
  2. x+y is in W whenever x and y are in W; and
  3. ax is in W whenever x is in W and a is any scalar.
symmetric matrix:
A matrix A is symmetric if it equals its transpose; i.e., A = AT.