Linear Algebra Definitions

characteristic polynomial

(for a matrix A) det (A-lambaI)

linear independence

(for a set of vectors) An indexed set {v1...vp} with the property that there exist weights c1...cp all equal to zero such that c1v1 +...+cpvp = 0

spanning set

(for a subspace H) any set {v1...vp} in H such that H = span {v1...vp}

change-of-coordinates matrix

(from a basis B to a basis C) a matrix that transforms B coordinate vectors into C coordinate vectors.

linear transformation

(from a vector space V into a vector space W): A rule T that assigns to each vector x in V a unique vector T(x) in W such that (i) T (u+v) = T(u) + T(v) (ii) T(cu) = cT(u) for all u in V and all scalars c

similar matrices

2 matrices A and B are similar if A = PBP^-1 for some invertible matrix P

one-to-one

A mapping T: R^n -> R^m is one-to-one if each b in R^m is the image of at most one x in R^n.

onto

A mapping T:R^n -> R^m is onto if each b in R^m is the image of at least one x in R^n.

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

eigenvalue

A scalar lambda such that Ax = lamba x has a solution for some nonzero vector x

subspace

A subspace of a vector space V is a subset H of V that has three properties: a. The zero vector of V is in H b. H is closed under vector addition. That is, for each u and v in H, the sum u+v is in H. c. H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector cu is in H.

basis

An indexed set B = {v1...vp} in V such that (i) B is a linearly independent set and (ii) the subspace spanned by B coincides with H, that is, H = span {v1...vp}

elementary matrix

An invertible matrix that results by performing one elementary row operation on an identity matrix

column space

The column space of an m x n matrix A is the set of all linear combinations of the columns of A. Col A = span {a1, a2,... an} where ai represents a column of A

adjugate

The matrix adj A formed from a square matrix A by replacing the (i,j) entry of A by the (i,j) cofactor, for all i and j, and then transposing the matrix

null space

The null space of an m x n matrix A is the set of all solutions to Ax = 0

inner product

The scalar u^Tv, usually written as u.v , where u and v are vectors in R^n. also called the dot product

row space

The set Row A of all linear combinations of the vectors formed from the rows of A; also denoted by Col A ^T

transpose

The transpose of a matrix A (of dimension m x n) is the n x m matrix whose columns are the corresponding rows of A

eigenvector

a NONZERO vector x such that Ax = lamba x for some scalar lambda

isomorphism

a one-to-one linear mapping from one vector space onto another

orthonormal set

a set of vectors is orthonormal if it is an orthogonal set of unit vectors (all of the vectors dotted with each other vector is 0)

orthogonal matrix

a square matrix U such that U^-1 = U^T. such a matrix has orthonormal columns. any square matrix with orthogonal columns is an orthogonal matrix.

standard matrix

for a linear transformation T: R^n -> R^m, there exists a unique matrix A such that T(x) = Ax for all x in R^n. This is called the standard matrix. A = [T(e1) ... T (en)]

dimension

the dimension of V, written as dim V, is the number of vectors in a basis for V. the dimension of the zero vector space {0} is 0.

rank

the dimension of the column space of A

kernel

the kernel of a linear transformation T: V -> Wis the set of x in V such that T(x) = 0

orthogonal projection

the orthogonal projection of y onto u is [(y.u)/(u.u)](u)