Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2014
Problem Set th4
Due on October 24
Problems preceded by * are harder and/or more involved
Problem 1. (25 points)
In assignment 2, we discretized the 1di
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2014
Problem Set 5
Due in class on November 5th
Problems preceded by * are harder and/or more involved
Problem 1. (15 points)
Let q1 , , qk be k orthonormal
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2014
Problem Set 5: Solutions
Problem 1.
T
First note that we can write Qk QT as k qi qi . Now we can use induction to do
k
i=1
the proof. As the base case,
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2015
Problem Set 1
Due in class on September 30th
Problems preceded by * are harder and/or more involved
Note: Some of the problems in this assignment (regar
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2014
Problem Set 1 st
Due in class on October 1
Problems preceded by * are harder and/or more involved
Problem 1. (10 points) Strang, problem 42 on page 30.
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2014
Problem Set 4: Solutions
Problem 1.
a) The LU factorization can be found by using Matlab command lu. To check
whether Matlab is using any pivoting we ca
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2014
Problem Set 6: Solutions
Problem 1.
a)
(i) The following steps lead us from C to B:
Bij = #shared actors between movies i and j
n
=
(1 if actor k is in
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2014
Problem Set 8: Solutions
Problem 1.
a) The characteristic polynomial is
p() = det(A I) = 2 26 + 104.
Solving p() = 0 for , the eigenvalues are
26 + 262
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra
with Applica1on to Engineering Computa1ons
CME200/ME300A
Gianluca Iaccarino
Linear Algebra
with Applica1on to Engineering Computa1ons
CME200/ME300A
Gianluca Iaccarino
Janlooka Yakkareeno
CME200/ME300A
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2016
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2015
Problem Set 8
Due on November 30th
Problems preceded by * are harder and/or more involved
Problem 1 (10 points)
(Same as Strangs problem 5.8 and 5.18, p
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2016
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2015
Problem Set 7
Due in class on November 18th
Problems preceded by * are harder and/or more involved
Problem 1. (25 points)
Consider the matrix
2 1 0
A =
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2016
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2015
Problem Set 5
Due in class on November 4th
Problems preceded by * are harder and/or more involved
Problem 1. (20 points) Strang, problems 9 and 11 on pa
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
CME 200 Assignment 2 Solutions
Stanford University
Problem 1a)
In one dimension with f (x) = 0 and boundary conditions T = 1 at x = 0 and T = 3 at
x = 4 we are solving the dierential equation
d2 T
= f (x),
dx2
with T (0) = 1 and T (1) = 4
The solution to
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2015
Workshop 3
Friday October 9th
Problem 1.
Prove that all bases of a subspace must have the same number of elements.
Proof : Suppose not. Let cfw_u1 , . .
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
CME 200/ME300A
G. Iaccarino
Fall 2015
Linear Algebra with Application
to Engineering Computation
Problem Set 4
Due: October 21, in class
No late assignments accepted
Issued: October 14, 2015
Important:
Give complete answers: Do not only give mathematical
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2015
Problem Set 2
Due in class on October 7th
Problems preceded by * are harder and/or more involved
Problem 1. (80 points)
In this assignment, we will solv
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2015
Problem Set 3
Due in class on October 14th
Problems preceded by * are harder and/or more involved
Problem 1. (25 points, 5 points for each part) Strang,
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2015
Problem Set 3: Solutions
Problem 1.
Remember that a set S is a linear subspace if for any two sequences x, y S
and any real number a, both x + y and ax
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2014
Problem Set 3: Solutions
Problem 1.
The condition number is given by P P 1 . Therefore we need to nd the norms
(both spectral and Frobenius) of P and P
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2014
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2014
Problem Set 2: Solutions
Problem 1.
a) In one dimension with f (x) = 0 and boundary conditions T = 0 at x = 0 and
T = 2 at x = 1 we are solving the dier
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with Application to
Engineering Computations
CME 200/ME 300A
G. Iaccarino
Autumn 2016
Problem Set 2
Problem Set 2
Due Oct. 12th, 2016
1. Problem #1: Heat Equation (50 points)
We are interested in solving the 1D heat equation numerically wit
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with Application to
Engineering Computations
CME 200/ME 300A
G. Iaccarino
Autumn 2016
Problem Set 1
Problem Set 1
Due Oct. 5th, 2016
1. Problem #1: Matrix Operations (20 points)
True or false? Prove or give a counterexample.
(a) If A2 is d
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with Application to
Engineering Computations
CME 200/ME 300A
G. Iaccarino
Autumn 2016
Problem Set 7
Problem Set 7
Due Dec. 5th, 2016
1. Problem #1 (20 points)
(a) (10 points) Find and ~x so that ~u = et ~x is a solution of
d~u
=
dt
4 3
0 1
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with Application to
Engineering Computations
CME 200/ME 300A
G. Iaccarino
Autumn 2016
Problem Set 3
Problem Set 3
Due Oct. 19th, 2016
1. Problem #1 (15 points)
Let P be an n n permutation matrix. Show that its condition number is 1 when com
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with Application to
Engineering Computations
CME 200/ME 300A
G. Iaccarino
Autumn 2016
Short Quiz 1
Short Quiz (not graded)
November 18th, 2016
Provide a short proof or a counterexample.
1. If A is a skew symmetric matrix, for any vector ~x,
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with Application to
Engineering Computations
CME 200/ME 300A
G. Iaccarino
Autumn 2016
Short Quiz 2
Short Quiz (not graded)
November 30th, 2016
Provide a short proof or a counterexample.
1. The eigenvalues of A and AT are the same.
2. The co
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with
Application to
Engineering Computations
CME200/ME300A
Gianluca Iaccarino
Fall 2015
Problem Set 1: Solutions
Problem 1.
(a) True. Assume that BA isnt invertible i.e., there is a nonzero x such that
BAx = 0. If Ax = 0, this will contrad
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with Application to
Engineering Computations
CME 200/ME 300A
G. Iaccarino
Autumn 2016
Problem Set 6
Problem Set 6
Due Nov. 16th, 2016
1. Problem #1 (65 points)
For this problem we will consider again the PageRank algorithm. Recall the linea
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with Application to
Engineering Computations
CME 200/ME 300A
G. Iaccarino
Autumn 2016
Problem Set 4
Problem Set 4
Due Oct. 26th, 2016
1. Problem #1 (40 points)
(a) (10 points) Use Matlab command A=rand(4) to generate 44 random matrices and
Linear Algebra with Application to Engineering Computations
CME 200

Fall 2015
Linear Algebra with Application to
Engineering Computations
CME 200/ME 300A
G. Iaccarino
Autumn 2016
Problem Set 5
Problem Set 5
Due Nov. 9th, 2016
1. Problem #1 (15 points)
Let Qk be an n k matrix (k n) with orthonormal columns columns ~q1 , . . . , ~qk