Math/Comp 128
Take-home Exam 1
Fall 2005
Instructions: The instructor is the only person you may consult with in
working on this exam. You may use your textbook, homework, calculator, and
class notes, and you may check answers using Matlab where appropria
Math/Comp 128, Homework 2
Due Sept. 22, 2005
1. Text, 2.1
2. Text, 2.2, part a only
3. Text, 4.1b,c,e
4. Text, 4.3 (this is a Matlab assignment). (You only need to apply your algorithm to matrices
in 4.1b,c,e). You may use the built-in Matlab svd function
Math/Comp 128 Homework 3
Due Sept. 29, 2005
1. Let Q be an n n unitary matrix. Show that Q
2
= 1 and Q
F
=
n.
2. Show that the product of two unitary matrices is unitary.
3. Let A be m n with m > n and assume that A is full rank. (Thus, rank
= n.)
(a) Use
Math/Comp 128
Homework 4
Due Oct. 6, 2005
1. Number 6.1 in text (pg. 47)
2. Number 6.4 in text (pg. 47)
3. Number 7.1a in text
4. Repeat exercise 3 on Homework 3 but using the reduced QR factorization
for A.
5. Program the classical Gram-Schmidt algorithm
Math 250NLA
Take-home Exam 1
Fall 2005
Instructions: The instructor is the only person you may consult with in
working on this exam. You may use your textbook, homework, calculator, and
class notes, and you may check answers using Matlab where appropriate
Math 128/ Comp 128
EXAM 2
Spring 2004
INSTRUCTIONS: Write all your answers in your blue book. Cross out anything you do not
want graded. No notes or textbooks are allowed. Calculators are permitted as long as they are not
mobile phones. No other electroni
Solutions for Homework Assignment 8
Math/Comp 128
23.1. We have that A is a nonsingular square matrix with its QR factorization
A = QR. Therefore, we have that A A = R Q QR = R R. Since A A is
hermitian positive denite, its Cholesky factorization is uniqu
Solutions for Homework Assignment 7
Math/Comp 128
3 2 17
1. We want to compute the LU factorization of the matrix A = 3 2 1.
1 1
1
3 2
17
1
00
4
18 .
First, we dene L1 = 1 1 0 and compute L1 A = U1 = 0
0 5/3 14/3
1/3 0 1
1
0
0
1
0 .
From this last matrix,
Solutions for Homework Assignment 6
Math/Comp 128 - Math 250NLA
1. Let A = U V be the SVD of A. We want to nd the solution to the least
squares problem
minx Ax b 2 ,
with the smallest 2-norm. Using the SVD we can rewrite the least squares
problem as follo
Solutions for Homework Assignment 4
Math/Comp 128
1. Since P is an orthogonal projector, we know that P = P 2 and P = P . We
want to show that I 2P is unitary. It is enough to show that (I 2P ) (I 2P ) =
I . We have that
(I 2P ) (I 2P ) = (I 2P )(I 2P ) =
Solutions for Homework Assignment 2
Math/Comp 128
1. The 2-norm and the Frobenius norm are invariant under unitary multiplication, that is, if Q is unitary, then QA 2 = A 2 and QA F = A F for
any matrix A. In particular, if A = I , we have that Q 2 = I 2
Solutions for Homework Assignment 2
Math/Comp 128
1. Assume the matrix A is upper triangular. Then its inverse, A1 , is upper
triangular. On the other hand, its adjoint matrix, A , is lower triangular. By
denition of unitary matrices, A1 = A . Then, we ha
Homework Assignment 1
Math/Comp 128
Due Sept. 15, 2005
1. Does the matrix
21
A= 4 2
1 1
satisfy the denition of a full-rank matrix? Justify your answer.
2. Text, 1.1, part a. The idea is that each step can be executed by pre or
post multiplying the curre