Adjectives/GENERAL TERMS!!! (Full semester)

Duke Bfitz - Spring 2024

An m x n matrix is square if…

m = n

The diagonal of a matrix refers to…

the (i, i) entries

A matrix is upper triangular if…

every entry below the diagonal is zero

A matrix is lower triangular if…

every entry above the diagonal is zero

A matrix is diagonal if…

every nondiagonal entry is zero

The diag(d1, . . . , dn) is…

the n × n diagonal matrix with diagonal d1, . . . , dn

Zero vectors and zero matrices are…

full of zeros

Nonzero

At least one entry in a matrix or vector is not zero

An n x n indentity matrix In

Has ones on the diagonal and zeros elsewhere

Standard basis {e1, . . . , en} of Rn

Are the columns of In

First pivot column

The first nonzero column

Rank one

every column is a multiple of the first pivot column

Orthogonal

Two vectors are _____ if their dot product is zero. This means they are at a right angle to each other.

Orthonormal

A set of vectors is _____ if all vectors in the set are mutually orthogonal and each of unit length.

Nonsingular (or Invertible)

A square matrix is ______ if it has an inverse. This means its determinant is not zero.

Eigenvalue; eigenvector

For a square matrix A, if there exists a vector x such that Ax = λx, then λ is an _____ and x is the corresponding _____.

Basis

A _____ of a vector space is a set of linearly independent vectors that span the entire vector space.

Rank

The _____ of a matrix is the maximum number of linearly independent rows or columns in the matrix.

Null space

The _____ of a matrix A is the set of all vectors x for which Ax = 0.

Row space

The _____ of a matrix A is the set of all possible linear combinations of its row vectors.

Column space

The _____ of a matrix A is the set of all possible linear combinations of its column vectors.

Linear transformation

A function between two vector spaces that preserves the operations of vector addition and scalar multiplication is…?

Kernel

The _____ of a linear transformation is the set of all vectors that map to the zero vector.

Trace

The _____ of a square matrix is the sum of the elements on its main diagonal.

Determinant

The _____ is a special number that can be calculated from a square matrix. Ex: = [(top left * bottom right) - (top right * bottom left)]

Diagonalization

The process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map is….

Algebraic Multiplicity

The _____ of an eigenvalue in a matrix is the number of times it appears as a root of the characteristic equation.

Geometric Multiplicity

The _____ of an eigenvalue is the dimension of the corresponding eigenspace, which is the null space of (A - λI), where A is the matrix, λ is the eigenvalue, and I is the identity matrix.

Spectral Factorization

_____ is a method used to factor a matrix using its spectral decomposition. It involves expressing a matrix as a product of other matrices that are functions of the original matrix’s eigenvalues and eigenvectors.

Gram-Schmidt Algorithm

The _____ is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space R^n. The process takes a finite, linearly independent set S = {v1, …, vk} and generates an orthogonal set S’ = {u1, …, uk} that spans the same k-dimensional subspace of R^n as S

Linear Independence

A set of vectors is said to be _____ if no vector in the set can be written as a linear combination of the others.

Dimension

In the context of vector spaces, the dimension is the maximum number of linearly independent vectors in the space.

The Four Fundamental Subspaces

These are the column space, row space, null space, and left null space of a matrix. If you have one, you can find all the others.

If the determinant of the matrix is zero, then the matrix is…?

singular

If the determinant of the matrix is not zero, then the matrix is…?

nonsingular

If the matrix has a row or column of zeros, it is _____ because the determinant will be zero.

singular

If two rows or columns of the matrix are identical or proportional (one is a scalar multiple of the other), the matrix is _____

singular