ECE 174 Homework # 2 Due Tuesday 10/28/2014
Reading
Immediately begin reading Sections 4.14.5. Particularly ponder the Summary of Rank chart on page
218. You can ignore examples 4.3.6, 4.4.6, 4.4.7, 4.5.2, and 4.5.3. (Later, but not now, we will read 4.6
ECE 174
Intro to Linear &
Nonlinear Optimization
Ken KreutzDelgado
ECE Department UCSD
Contact Information Fall 2014
! Course Website
Accessible from http:/dsp.ucsd.edu/~kreutz
! Course Time & Location
TuTh  56:20pm, Room CSB001
! Instructor
Prof.
ECE 174 Sample Midterm Question Solutions
1. The denitions can be found in the lecture notes and textbook. Note that this list is
not exhaustive and other denitions can also be asked for on the exam (such as the
denitions of eld; onto; onetoone; rank; e
ECE 174 FINAL EXAM FALL 2014 SOLUTIONS Total
[10 pts total] 1. Let matrix A denote a linear mapping between two complex weighted inner product spaces as follows.
Let the domain be C3 with inner product weighting matrix,
4.80
1 + 2 j 0.01 + 0.02 j
.
9.76
2
ECE 174 Computer Assignment #2
GLOBAL POSITIONING SYSTEM (GPS) ALGORITHM
Overview
By utilizing measurements of the socalled pseudorange between an object and each of
several earth orbiting GPS satellites, the object can be very accurately located in spac
ECE 174
Supplemental Solutions to Homework 2
The material presented below supplements the solutions which you can nd in
the textbooks Solutions Manual (which all of you should have).
Meyer 4.1.1. Is the subset of Rn a vector subspace? Since Rn is a vector
ECE 174 COMPUTER ASSIGNMENTS
General Instructions
Computer assignments must be completed by the indicated due dates. Matlab code plus
a report must be turned in for each separate assignment by the designated date.
The computer assignment reports will be r
Sample ECE 174 Final Questions
No solutions are provided
NOTE: This is a superset of the sample midterm questions. This is because the nal exam is comprehensive
and will ask questions about the material covered during the entire quarter. In particular, on
ECE 174 MIDTERM SOLUTIONS FALL 2014
1
1. (a) We are given that the data yi x, i = 1, , N , provides N noisy measurements of the unknown
(scalar) distance x = d. This can be written as,
1
y1
1
y1
. .
.
. . x , or y Ax for y = . and A = . .
.
.
.
ECE 174 Midterm
1.

Fall 2014
lV Earthorbiting sate\itgr provide independent noisy measurements,
U.; =
1, . . . ,lV, of the distance,)glf from the Earth of a dg
space probe exploring
the outer reaches of the Solar System. Given
least squares estimate of
ECE 174
Supplemental Solutions to Homework 2
The material presented below supplements the solutions which you can nd in
the textbooks Solutions Manual (which all of you should have).
Meyer 4.1.1. Is the subset of Rn a vector subspace? Since Rn is a vector
ECE 174 Computer Assignment #1
LEAST SQUARES AUDIO AND SPEECH COMPRESSION
LINEAR PREDICTIVE CODING (LPC)
Background on LPC Lossy Compression of Speech Signals
Speech, and other audio, signals represented by sample data Y N = cfw_y(n), n = 1, 2, , N ,
are
ECE 174 Homework # 4 Solutions
1. The two situations correspond to the mutually exclusive cases of a) b is in the range
of A, and b) nonzero b is not in the range of A.1
2. Meyer Problem 4.6.7. Let the data pairs be given by (xk , yk ), k = 1, , m = 11.
(
ECE 174 Homework # 5 Due Thursday, December 3, 2015
Comments and Reading
This is the last homework assignment. Although it is due on Thursday of the 10th week of the
quarter (i.e., on the very last lecture of the quarter), you should start look at these p
ECE 174  Homework # 5 Solutions
1. The MLE, xMLE , is determined as
2
xMLE = arg min cfw_ ln p(y x) = arg min
x
x
i=1
1
(y x)2 = arg min y Ax
2 i
x
i
where
y=
y1
y2
,
A=
1
1
,
W =
w1 0
0 w2
=
1
2
1
0
0
1
2
2
2
W
,
.
We have that the adjoint operator is
ECE 174 Homework # 4 Due Thursday 11/19/15
NOTE: Do this homework set as soon as possible. Otherwise you will be short of time at
the end of the quarter because Homework 5 and the second computer assignment are both
due on the last lecture of the quarter.
ECE 174 Homework # 3 Due Thursday, 11/5/2015
There are ten (10) questions on this homework assignment. Remember, the Solutions
Manual is provided with the text. Errata for the textbook are available at the website:
http:/MatrixAnalysis.com
Reading From Ch
ECE 174 Homework # 2 Due Tuesday 10/20/2015
READING
Immediately begin reading Sections 4.14.5. Particularly ponder the Summary of Rank chart on page
218. You can ignore examples 4.3.6, 4.4.6, 4.4.7, 4.5.2, and 4.5.3. (Later, but not now, we will read 4.6
ECE 174 Fall 2015
Supplemental Solutions to Homework 3
1. It is obvious that the rank of the matrix is 2 (as the two rows and the
rst two columns are linearly independent). The two linearly independent
rows span the row space (i.e., they form a basis for
ECE 174 Homework # 1 Due Thursday 10/1/2015
Assumed Background
Students are assumed to have had a prior course in Linear Algebra, and therefore to already
know relevant material presented in Chapters 13 of the assigned textbook (see the reading
requireme
ECE 174 Homework # 1 Due Thursday 10/9/2014
Assumed Background
Students are assumed to have had a prior course in Linear Algebra, and therefore to already
know relevant material presented in Chapters 13 of the assigned textbook (see the reading
requireme
Review Material
Chapters 13 of Matrix Analyis
Textbook
Textbook
Example 1.2.1
Textbook
Example 1.3.1
Textbook
Example 2.1.1
Textbook
Example 2.1.2
Textbook
Example 2.2.2
If such
relationships
exist, then the
columns
are said to be
"linearly
dependent."
O
ECE 174 Homework # 4 Solutions
1. The two situations correspond to the mutually exclusive cases of a) b is in the range
of A, and b) nonzero b is not in the range of A.1
2. Meyer Problem 4.6.7. Let the data pairs be given by (xk , yk ), k = 1, , m = 11.
(
ECE 174 Fall 2016
Supplemental Solutions to Homework 3
1. It is obvious that the rank of the matrix is 2 (as the two rows and the
first two columns are linearly independent). The two linearly independent
rows span the row space (i.e., they form a basis fo
ECE 174 Midterm Fall 2016 100 Points Total
Solutions
(15 points) 1. V and W are two companion subpaces of a vector space X of dimension n > 1. For each set
listed below, determine if there exists a projection operator that projects onto that set. For
each
ECE 174 Midterm Question Solutions  Fall 2011
Sujitha Martin
Professor KreutzDelgado
November 18, 2011
The following are solutions to midterm questions from Fall 2011. If you find any mistakes
in the solution, please contact Professor KreutzDelgado or
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Solutions ECE 174 Midterm Spring 2007
1. Let V be a real m n matrix that maps from the real vector space X to the vector
space Y. Both X and Y are cartesian spaces with the standard inner product.
Assume that the columns of V form a basis for the subspace
1
ECE 174 Midterm Fall 2010 Solutions
1. (35 pts) Let the m n matrix A represent a linear mapping between two complex Hilbert spaces
X and Y with innerproduct weighting matrices given by and W respectively. (You can
assume that all vectors are represente
ECE 174 Midterm
1.

Fall 2014
lV Earthorbiting sate\itgr provide independent noisy measurements,
U.; =
1, . . . ,lV, of the distance,)glf from the Earth of a dg
space probe exploring
the outer reaches of the Solar System. Given
least squares estimate of
Pseudoinverse & Orthogonal Projection Operators
ECE 174 Linear & Nonlinear Optimization
Ken KreutzDelgado
ECE Department, UC San Diego
Ken KreutzDelgado (UC San Diego)
ECE 174
Fall 2016
1 / 48
The Four Fundamental Subspaces of a Linear Operator
For a li
Vector Derivatives, Gradients, and Generalized Gradient
Descent Algorithms
ECE 174 Introduction to Linear & Nonlinear Optimization
Ken KreutzDelgado
ECE Department, UC San Diego
December 4, 2016
Ken KreutzDelgado (UC San Diego)
ECE 174
December 4, 2016
ECE 174  Homework # 5 Solutions
1. (a) The MLE, x
bMLE , is determined as
x
bMLE
2
X
1
= arg min cfw_ ln p(y x) = arg min
(y x)2 = arg min ky Axk2W ,
2 i
x
x
x
i=1 i
where
y
y= 1 ,
y2
1
A=
,
1
1
12
w1 0
W =
=
0 w2
0
0
1
22
!
.
We have that the adjoi