Linear Algebra Final Exam terms

Transpose of a matrix

an nxm matrix A^T whose columns are the corresponding rows of the mxn matrix A.

Inverse of a matrix

an nxn matrix A^-1 such that AA^-1 = A^-1A = In.

Elementary Matrix

A matrix which differs from the identity matrix by a single elementary row operation.

Linear Transformation

A function that maps vectors in one vector space to another while preserving vector addition and scalar multiplication.

Eigenvalue of a matrix

there is a nontrivial solution x of Ax = λx; such an x is called an eigenvector corresponding to λ.

Eigenvector of a matrix

is a nonzero vector x such that Ax=λx for some scalar λ.

Eigenspace of a matrix

the set of all solutions of Ax = λx, where λ is an eigenvalue of A.

Characteristic Polynomial

det(A-λI)

Column Space of a matrix

is the set Col A of all linear combinations of the columns of A.

Null space of a matrix

is the set Nul A of all solutions of the homogeneous equation Ax=0.

Subspace

any set H in R^n that has 3 properties: 1) the zero vector is in H. 2) For each u and v in H, the sum u+v is in H. 3) For each u in H and each scalar c, the vector cu is in H.

Basis of a subspace

is a linearly independent set in H that spans H.

Dimension of a vector space

denoted by dim H, is the number of vectors in any basis for H.

Rank of a matrix

denoted by rank A, is the dimension of the column space of A.

Nullity of a matrix

is the dimension of the null space of A.

Diagonalizable matrix

a matrix that can be written in factored form as PDP^-1, where D is a diagonal matrix and P is an invertible matrix.

Norm of a vector

The scalar ||v|| = sqrt (v*v) = sqrt(v,v).

Distance between 2 vectors (dist (u,v))

the length of the vector u-v

orthogonal vectors

if u *v = 0.

orthogonal complement of a subspace

the set W^⊥ of all vectors orthogonal to W.

orthogonal basis

is a basis for W that is also an orthogonal set.

orthonormal basis

a basis that is an orthogonal set of unit vectors.

symmetric matrix

a matrix A such that A^T = A.

orthogonal matrix

a square invertible matrix U such that U^-1 = U^T.

orthogonally diagonalizable matrix

if there are an orthogonal matrix P (with P^-1 = P^T) and a diagonal matrix D such that A = PDP^T = PDP^-1.

spectral decomposition of a symmetric matrix

it breaks up A into pieces determined by the eigenvalues of A.

A = λ1 u1 u1^T + λ2 u2 u2^T + … + λn un un^T.

colinear vectors

if they lie on the same line or parallel lines.

linear combination of a set of vectors

the vector y is defined by

y = c1 v1 + … + cp vp.

linear independence of a set of vectors

if the vector equation has only the trivial solution

x1 v1 +…+ xp vp = 0.

range of a linear transformation

the set of all vectors of the form T(x) for some x in the domain of T.

standard matrix of a linear transformation

the matrix A such that T(x) = Ax for all x in the domain of T.

onto linear transformation

if each b in R^m is the image of at least one x in R^n.

one-to-one linear transformation

if each b in R^m is the image of at most one x in R^n.

span of a set of vectors

the set of all linear combinations of v1, …, vp.

domain

the set of all vectors x for which T(x) is defined.

codomain

the set R^m that contains the range of T.

linear dependence of a set of vectors

if there exists weights c1, …, cp not all zero, that is, the vector equation has the nontrivial solution.