Discussion 2
Lillian Ratli
6 September 2013
1
1.1
Review of Lecture
Functions
Denition 1.1
A function f : X Y is map (relation) from X into Y such that f assigns a unique f (x) Y .
Denition 1.2
A function f : X Y is injective (onetoone) if for all x1 , x2
EE221A Linear System Theory
Problem Set 2
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2011
Issued 9/8; Due 9/16
All answers must be justied.
Problem 1: Linearity. Are the following maps A linear?
(a) A(
Problem Set 2
EE221a: Linear Systems Theory
Prof. S.S. Sastry
Fall 2013
Issued: September 10th, 2013.
Due: September 24th, 2013.
1. This is taken from Exercise 3, pg. 28 of the Notes for a Second Course in Linear Systems, by Desoer.
Since you may not have
EE221A Linear System Theory
Problem Set 2
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
Issued 9/18; Due 9/27
Problem 1: Useful properties of eigenvalues. Let A Rnm , B Rmn and let n m. Observe that
EE221A Problem Set 2 Solutions - Fall 2011
Problem 1. Linearity.
a) Linear: A(u(t) + v (t) = u(t) + v (t) = A(u(t) + A(v (t)
b) Linear:
t
e (u(t ) + v (t )d =
A(u(t) + v (t) =
0
t
e u(t )d +
0
t
e u(t )d
0
= A(u(t) + A(v (t)
c) Linear:
2
s
2
A(a1 s + b1 s
EE 221: Linear Systems HW #5 Solutions Sam Burden Oct 15, 2010
1
Note these solutions are overly terse and leave out some details; in your solutions, you should strive for greater clarity and rigor. Exercise 1. Time invariance is straightforward to verify
EE221A Linear System Theory
Problem Set 6
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
Issued 11/6; Due 11/15
Problem 1: Sti Dierential Equations.
In the simulation of several engineering systems we
EE221A Linear System Theory Problem Set 5
Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2010 Issued 10/8; Due 10/15
You may pick up solutions from us on 10/14, 10/15 or 10/18, after you hand in your homew
EE221A Linear System Theory
Final Exam
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2010
12/14/10, 3-6pm
Your answers must be supported by analysis, proof, or counterexample.
There are 9 questions: Pleas
Sam Burden Linear Systems (EE 221) Midterm Review Oct 15, 2010
1
Outline of Course
1. Preliminaries: f : X Y , F , (V, F ), cfw_bj n 1 (a) functions: injective, surjective, bijective, left & right inverse (b) eld, ring (c) vector space, subspace (d) linea
EE221A Linear System Theory
Midterm Test
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
10/16/07, 9.30-11.00am
Your answers must be supported by analysis, proof, or counterexample.
There are 6 questio
10/7/11
EE221A Section 7
1
Practice midterm
Problem 1. Injectivity and surjectivity
a) Suppose that T : V W is an injective, linear map, and that cfw_v1 , . . . , vn
is a linearly independent set in V . Prove that cfw_T (v1 ), T (v2 ), . . . , T (vn ) is
11/04/11
EE221A Section 10
1
Direct sum of subspaces
Exercise 1. Show that if V = V1 V2 Vn , then Vi Vj = cfw_ for i = j .
Exercise 2. Let M and N be two subspaces of V . Let cfw_m1 , . . . , mp be a basis
for M , and cfw_n1 , . . . , nk be a basis for
9/2/11
EE221A Section 2
1
Fields
1. Show that the set cfw_0, 1, with multiplication dened as binary AND and addition
dened as binary XOR, is a eld.
(AND)
0
1
+ (XOR) 0 1
0
01
1
10
01
00
01
2. Show that F, 0 = 0 = 0.
2
Vector Spaces
1. Does C form a vecto
10/28/11
EE221A Section 9
1
Administrivia
Midterm avg 28, median 29, std dev 4.9 (out of 34)
HW6 is out, due next Fri (Nov 4)
2
Cayley-Hamilton Theorem
Recall:
Characteristic polynomial of A: A (s) := det (sI A) = sn + d1 sn1 + + dn
Characteristic equat
10/21/11
EE221A Section 8
1
Administrivia
Midterms still being graded.
HW5 is out, due next Thurs (Oct 27)
GSI oce hours Mon Oct 24th time change to 1 PM (still in 504 Cory)
2
Dynamical systems
Exercise 1. Show that the following system is time invaria
9/16/11
EE221A Section 4
1
Change of basis
Exercise 1. [LN3, p. 10]
Let A : R3 R3 be a linear map. Consider
1
B = cfw_b1 , b2 , b3 = 0 ,
0
1
C = cfw_c1 , c2 , c3 = 1 ,
0
0
1 ,
0
0
1 ,
1
0
0 ,
1
1
0 .
1
Clearly B and C are bases for R3 . Suppose A
9/23/11
EE221A Section 5
1
Norms
Exercise 1. Prove that x Rn , x
x
1
n x
Exercise 2. In R2 , sketch the unit sphere B =
p = . What about 0 < p < 1?
2
x: x
p
=1
for p = 1, p = 2,
Complete (Banach) Spaces
Exercise 3. Let X be the space of real-valued contin
9/30/11
EE221A Section 6
1
Singular Value Decomposition
Exercise 1. Show that the eigenvalues of a Hermitian matrix are all real.
Exercise 2. Show that AA , for A Cmn , is positive semidenite.
Exercise 3. Consider a real unitary matrix U R33 . Give a geom
9/9/11
EE221A Section 3
1
1.1
Functions, linear maps
Solutions to linear equations
Theorem. (range and nullspace of linear operators) [LN3 p. 4]
Consider A : U V with (U, F ), (V, F ) linear spaces. Let b V . Then:
a) A(u) = b has at least one solution b
EE221A Linear System Theory
Problem Set 1
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
Issued 9/4; Due 9/13
Problem 1: Fields.
(a) Dene addition and multiplication on cfw_0, 1 to form a eld. Show th
EE221A Linear System Theory
Problem Set 3
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
Issued 9/27; Due 10/4
Problem 1: Dynamical systems, time invariance.
Suppose that the output of a system is rep
EE221A Linear System Theory
Problem Set 4
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2006
Issued 10/9; Due 10/18
Dear 221A folks: Some problems like the ones below may be included on the midterm on Oct
EE221A Linear System Theory
Problem Set 5
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
Issued 10/23; Due 11/1
Problem 1.
Suppose A Cnn is such that det(A) = 0. Is det(eA ) = 0? Explain why or why no
EE221A Linear System Theory
Problem Set 7
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
Issued 11/15; Due 11/27
Problem 1. Show that
A0
c0
,
b
0
where A Rnn , b Rn , cT Rn is completely controllable
EE221A Linear System Theory
Problem Set 8
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
Issued 11/27; Due 12/6
Problem 1: State vs. Output Feedback.
Consider the plant described by:
X
y
where
A=
0
7
EE221A Linear System Theory
Problem Set 9
Professor C. Tomlin
Department of Electrical Engineering and Computer Sciences, UC Berkeley
Fall 2007
Issued 12/4; Due 12/15
221A folks: Homework 9 is due at the nal exam. However, if youd like to hand in your hom