### 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 {v_{1}, ..., v_{k}} in W such that:- {v
_{1}, ..., v_{k}} is linearly independent; and - {v
_{1}, ..., v_{k}} spans W.

- {v
**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:- all rows that consist entirely of zeros are grouped together at the bottom of the matrix; and
- 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 v_{1}, ..., v_{k}if there exist scalars a_{1}, ..., a_{k}such that v = a_{1}v_{1}+ ...+ a_{k}v_{k}.**linear dependence relation**for a set of vectors:A

*linear dependence relation*for the set of vectors {v_{1}, ..., v_{k}} is an equation of the form a_{1}v_{1}+ ...+ a_{k}v_{k}= 0, where not all the scalars a_{1}, ..., a_{k}are zero.**linearly dependent**set of vectors:- The set of vectors {v
_{1}, ..., v_{k}} is*linearly dependent*if the equation a_{1}v_{1}+ ...+ a_{k}v_{k}= 0 has a solution where not all the scalars a_{1}, ..., a_{k}are zero (i.e., if {v_{1}, ..., v_{k}} satisfies a linear dependence relation). **linearly independent**set of vectors:The set of vectors {v

_{1}, ..., v_{k}} is*linearly independent*if the only solution to the equation a_{1}v_{1}+ ...+ a_{k}v_{k}= 0 is the solution where all the scalars a_{1}, ..., a_{k}are zero. (i.e., if {v_{1}, ..., v_{k}} 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:- T(u+v) = T(u) + T(v) for all vectors u and v in V; and
- 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**R**^{n}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**R**^{n}is the set of all vectors v in**R**^{n}such that v is orthogonal to every vector in S. **orthogonal set**of vectors:A set of vectors in

**R**^{n}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}= A^{T}. **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
**R**^{n}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.

- The
**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:- the matrix is in row echelon form;
- the first nonzero entry in each nonzero row is the number 1; and
- 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

^{-1}AS = 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 {v_{1}, ..., v_{k}} is the subspace V consisting of all linear combinations of v_{1}, ..., v_{k}. One also says that the subspace V is*spanned*by the set of vectors {v_{1}, ..., v_{k}} and that this set of vectors*spans*V.**subspace**:A subset W of

**R**^{n}is a*subspace*of**R**^{n}if:- the zero vector is in W;
- x+y is in W whenever x and y are in W; and
- 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 = A^{T}.