Linear Algebra Final Review

Linear combination of a set of vectors

The linear combination of the vectors V1,V2, …, Vn with scalars C1, C2, …, Cn is the vector

C1V1 + C2V2 + … + CnVn

The scalars C1, C2, …, Cn are called the weights of the linear combination

Span of a set of vectors

The span of a set of vectors V1,V2, …, Vn is the set of all linear combinations that can be formed from the vectors.

Alternatively, if A = [V1 V2 … Vn], then the span of the vectors consists of all vectors b for which the equation Ax = b is consistent.

Eigenvalue/Eigenvector pair for a matrix A

Given a square n x n matrix A, we say that a nonzero vector V is an eigenvector of A if there is a scalar λ such that

Av = λv

The scalar λ is called the eigenvalue associated to the eigenvector V

Similarity of matrices

We say that A is similar to B if there is an invertible matrix P such that A = PBP^-1

Diagonalizable matrix

We say that the matrix A is diagonalizable if there is a diagonal matrix D and invertible matrix P such that

A = PDP^-1

Orthogonal vectors

We say that vectors V and W are orthogonal if V ⋅ W = 0

Orthogonal matrix

A square m x m matrix Q whose columns form an orthonormal basis for R^m is called orthogonal