Exam 3 Linear Algebra

If the matrix A has orthogonal columns, then what is A^T A

A^TA would be a diagonalizable matrix

If A = QR is a QR factorization of A, then…

A^TA = R^TR

if x hat and y hat are least squares solution of Ax = b, then what does x hat - y hat belong to?

(x hat - y hat) belong to the Nul(A)

if U is a 3×2 matrix with orthonormal columns, then for every y vector in Col(U) ….

we have y vector = U*U^T*y

how do you diagonalize a matrix?

1. find eigenvalues that make up your diagonal matrix D (along the diagonal) det(A-λI) = 0

2. find eigenvectors that make up your P matrix by plugging in your λs into your (A-λI) = 0

3. check if eigenvectors are linearly independent.

4. solve for P^-1 based on it’s size and check if P^-1*P = I

what makes a matrix diagonalizable?

An n x n matrix is diagonalizable if it has n linearly ind. eigenvectors.

r/s btwn (Row A) perp….

(Row A)⟂ = Nul(A)

r/s btwn Col(A) perp…

(Col A)⟂ = Nul (A)^T

a vector x is in Nul(A) if and only if

• Ax = 0.

• this means that x is orthogonal to each row of A

• Row A is orthogonal complement to Nul A

• Dim (Row A) + Dim (Nul A) = N

  • the ambient dimension

Nul(A) is the orthogonal complement to

(Row A)

Nul (A^T) is the orthogonal complement to

(Col A)

What is Col(A) orthogonal complement and how?

Nul(A^T)

What is Nul(A)’s orthogonal complement and how?

Col(A)

The column space of A can also be referred to as

the row space of A^T

if vectors are all orthogonal and all zero… are they lin dep? or lin. ind?

linearly independent

how do you find a vector in a set of vectors that are orthogonal to all of them

transpose the vectors and create a matrix and solve for Ax = 0. the Nul of this will be the third matrix

how do you find the expansion of a vector s in an orthogonal basis

s = c₁u₁ + … c_pu_p

where c_p = ((s * u_p)/(u_p*u_p) * u_p…

orthogonal matrix

a square matrix whose columns are orthonormal

When does a matrix U have orthonormal columns?

An m x n matrix U has orthonormal columns iff U^TU = I_n

• does not apply when n > m

• orthonormal: columns have unit length (aka they are of length 1)

name for matrix with orthogonal columns

there is none!