Math 102 - Homework 7 (selected problems)
David Lipshutz Problem 1. (Strang, 5.2: #4) If a 3 by 3 upper triangular matrix has diagonal entries 1, 2, 7, how do you know it can be diagonalized? What is
MATH102 Midterm 2 solutions
Roman Kitsela
December 5, 2016
Question 1.
Proof. By the Rank-Nullity Theorem (rank(A)+dim(N ull(A) = Number of columns of A):
r + dim(N ull(A) = n = dim(N ull(A) = n r
We
MATH 102: PROBLEM SET 4
DUE AT 16:00 ON FRIDAY, OCTOBER 27
(1) Find the line of best fit through the points (2, 3), (3, 4), (4, 5), (5, 6).
(2) Let V be the vector space of continuous functions f : [0
Quiz 1 Solutions
1. (a) Adding R1 to 3R2 , we get the equation (1 3a)y = 5. Thus the system of equations fails
to have a unique solution if and only if 1 3a = 0, i.e. if and only if a = 1 .
3
(b) If a
MATH 102: PROBLEM SET 2
DUE AT 16:00 ON FRIDAY, OCTOBER 13
(1) Let V be a two-dimensional
vector space, let cfw_e1 , e2 be a basis in V , let
11 12
A=
be a 2 2 matrix, and define a function
21 22
h,
MATH 102: PROBLEM SET 5
DUE AT 16:00 ON MONDAY, NOVEMBER 13
(1) Let V be a 4-dimensional vector space, and let cfw_v1 , v2 , v3 , v4 be a basis
in V . Let A be the linear operator on V defined by
Av1
MATH 102: PROBLEM SET 6
DUE AT 16:00 ON MONDAY, NOVEMBER 20
Let V be a finite-dimensional Hilbert space.1 By solving the following problems,
you will verify some basic properties of selfadjoint and un
MATH102 Midterm 1 solutions
Roman Kitsela
December 5, 2016
Question 1.
Proof. Let B be a real 2n 2n matrix partitioned as:
B11 B12
B=
B21 B22
We calculate the product AB:
A11 A12 B11 B12
A11 B11 + A
Math 102, Winter 2015
Preliminary Definitions
Prof. Guershon Harel
Jan. 5, 2015
Remark: This document gives definitions of basic terms that will be essential in this class. These
terms were defined an
Math 102 - Winter 2013 - Midterm II
Name:
Student ID:
Section time:
Instructions:
Please print your name, student ID and section time.
During the test, you may not use books, calculators or telephones
Math 102 - Winter 2013 - Final Exam
Name:
Student ID:
Section time:
Instructions:
Please print your name, student ID and section time.
During the test, you may not use books, calculators or telephones
Math 102 - Winter 2013 - Midterm I
Name:
Student ID:
Section time:
Instructions:
Please print your name, student ID and section time.
During the test, you may not use books, calculators or telephones.
Math 102 - Winter 2013 - Midterm I
Name:
Student ID:
Section time:
Instructions:
Please print your name, student ID and section time.
During the test, you may not use books, calculators or telephones.
Math 102 - Winter 2013 - Midterm II
Name:
Student ID:
Section time:
Instructions:
Please print your name, student ID and section time.
During the test, you may not use books, calculators or telephones
Math 102 - Winter 2013 - Final Exam
Problem 1.
Consider the matrix
1
1
2
1
A=
1 1 .
2 1
(i)
(ii)
(iii)
(iv)
Find
Find
Find
Find
the
the
the
the
left inverse of A.
matrix of the projection onto the col
MATH 102: PROBLEM SET 3
DUE AT 16:00 ON FRIDAY, OCTOBER 20
(1) Let (V, h, i) be a Euclidean space. Prove the Parallelogram Law: for
any v, w V , we have
kv + wk2 + kv wk2 = 2kvk2 + 2kwk2 .
(2) Let V =
MATH 102: PROBLEM SET 7
DUE AT 16:00 ON MONDAY, NOVEMBER 27
(1) Let V be a finite-dimensional Hilbert space, and let W be a subspace of V .
Define the orthogonal complement of W to be
W = cfw_v V : hv
MATH 102: PROBLEM SET 1
DUE AT 16:00 ON FRIDAY, OCTOBER 6
(1) Prove that every vector space V contains a unique zero vector 0. Thus it
is correct to refer to 0 as the zero vector of a given vector spa
Math 102
Applied Linear Algebra
Summer Session II 2015
Quiz III
You may not use any reference materials for this quiz. All work must be your own.
0. What is your seat number?
1. Let A be an m n matrix
Math 102
Applied Linear Algebra
Summer Session II 2015
Quiz II
You may not use any reference materials for this quiz. All work must be your own.
0. What is your seat number?
1. (a) Which of the follow
Math 102
Applied Linear Algebra
Summer Session II 2015
Quiz IV
You may not use any reference materials for this quiz. All work must be your own.
0. What is your seat number?
y
-2
1. Consider the follo
Math 102
Applied Linear Algebra
Summer Session II 2015
Quiz I
You may not use any reference materials for this quiz. All work must be your own.
1. Consider the system of equations below. All numbers t
Math 102, Winter 2017
Midterm Review
Chapter 1. Matrices and Gaussian Elimination
(
x1 + 2x2 = 2,
The
1. Different forms of a system of linear equations. Example:
3x1 + 4x2 = 4.
1
2
2
vector form
W5 34; , (D
\) 3 2 )
0 3
L) 3 l t Z t H 3, . 2M): 2 Ab
2 3 2 3 3 \
' S I q I ~ 5 2 "' +1 1 1
U) 2 _ ' Z l 1 q 1
4 2 |
T 3 ~50 +0 -_- _1-'
l 2 Q6 '22 O o (5
5 2
+ R own/Hows [8431c
\
Math 102 HW 6
Exercise 1.
(1) We want to describe how any permutation (1, n) can be written as a
product of transpositions. Well denote the transposition which switches i and
j by i,j . If is the iden
Please check the website for your room assignment!
General remarks:
You will be asked to write down your name and student ID on the front page.
You have 50 minutes to work on the exam.
No books, calcu
MATH 102 BONUS HOMEWORK ASSIGNMENT
Due Friday, December 1st, 2017 before the lecture.
Handwritten submissions only.
Completing the following bonus exercise is completely voluntary. The points of the b
Math 102 HW 4
Exercise 1.
(a) To reduce fractions of complex numbers, multiply both the top and the
bottom by the conjugate of the bottom:
2i
i
+
3 + 4i 3 4i
3 + 4i
i
3 4i
2i
+
=
3 + 4i 3 + 4i
3 4i 3
MATH 102 HOMEWORK ASSIGNMENT 7
Due Friday, November 17th, 2017 before the lecture.
Handwritten submissions only.
Exercise 1 (4 points).
We conduct Gaussian elimination and the LU decomposition over th
MATH 102 HOMEWORK ASSIGNMENT 7
Due December 1st, 2017 before the lecture.
Handwritten submissions only.
Exercise 1 (4 points).
Let A, B Rnn . Prove the following:
(1) det(Ak ) = det(A)k for k N.
(2) d