matrix studying

column space

the set of all linear combinations of the columns of A

basis

a basis for a subspace H of Rn is a linearly independent set in H that spans H

• the minimum number of vectors you need to know to know the whole spaces

null space

the set of all solutions to Ax = 0

nonhomogeneous

a system of linear equations is non-homogeneous if it can be written as Ax = b where b doesn’t equal 0

homogeneous

if a system of linear equations can be written in the form Ax = 0 (A is a mxn matrix) (x is a column of unknowns) ( 0 is the 0 vector)

commute

if A and B are defined and AB=BA, we say A and B commute

invertible matrix

a nxn matrix A is invertible if there is a nxn matrix C such that AC = I_n = CA

C is an inverse of A and denoted A^-1 = C

singular matrix

a matrix that is not invertible

similar matrix

let a and b be nxn matrices. we say a is similar to be if there is an invertible matrix P such that

B=P^-1AP

• have same eigenvalues

diagonalizable

a nxn matrix A is diagnolizable if A has n linearly independent eigenvectors

• A is diagonalizable only if there are enough eigenvectors to form a basis of Rn

• the diagonal entries of D are eigenvalues

• a nxn matrix with n distinct eigenvalues is diagonalizable

rank = # of pivot columns

a matrix is invertible if its determinant doesn’t equal 0

orthogonal

two vectors are orthogonal if their dot products equal 0

orthogonal compliment

the set of all vectors orthagonal to the subspace W

• W perp

(colA)^perp = NullA^T

orthogonal projection

x onto the line through u

xu/uu

QR factorization

if A is an mxn matrix with linearly independent columns then A can be factored as A=QR where Q is an mxn matrix whose columns form an orthonormal basis for ColA and R is an nxn upper triangular invertible matrix with positive nonzero entries on the diagonal

symmetric matrix

A^T=A

orthogonally diagonalizable

A nxn matrix A is said to be orthogonally diagonalizable if there exists an orthogonal matrix P(P^-1=P^T) and a diagonal matrix D such that

A = PDP^-1 = PDP^T

quadratic form

a function Q defined on Rn such that Q(x) = x^TAx where A is a nxn symmetric matrix

• outputs a scalar