Linear Algebra Exam 4

Similarity

refers to the relationship between two matrices if one can be transformed into the other by a change of basis. Two matrices A and B are similar if there exists an invertible matrix P such that B = P^(-1)AP.

eigenvalue

a scalar that indicates how much a corresponding eigenvector is stretched or compressed during a linear transformation represented by a matrix.

eigenvector

a non-zero vector that changes by a scalar factor during a linear transformation represented by a matrix.

equation for similarity

If A and B are similar, then there exists an invertible matrix P such that B = P^(-1)AP.

Diagonalization theorem

An nxn matrix A is diagonalizable iff A has n linearly independent eigenvectors.

Not diagonalized

means that a matrix cannot be expressed as a product of a diagonal matrix and an invertible matrix, often due to insufficient independent eigenvectors. Not in the span Rn

Properties of Inner Product

A function that maps two vectors to a scalar, satisfying linearity, conjugate symmetry, and positive-definiteness.

Length of a vector

Norm - calculated by the sqrt(v²*v²)

unit vector

A vector with a length of one, often used to indicate direction in a vector space.

Distance Rn

The shortest path between two points in Rn, calculated using Norm (u - v)/Norm

Orthogonal components

refer to vectors that are perpendicular to each other in a vector space, typically used to simplify vector calculations and decompositions.

orthogonal basis

A basis where all vectors are mutually orthogonal.

Orthogonal Projection

Y=(yu/uu)u

To show that z is orthogonal to u

z*u=0 and also z= y-Y

best approximation theorem (BAT)

States that the orthogonal projection of a vector onto a subspace is the best approximation to that vector from the subspace.

Gram Schmidt Process

is an algorithm for producing an orthogonal basis for any non-zero subspace of Rn

Orthogonal Basis

Vectors/ Norm of those vectors

Least Squares

1. Find ATA and Find ATb