Mathematics 222 Calculus III Assignment 2 1. For the following power series, find (a) the radius of convergence (b) the interval of convergence, discussing the endpoint convergence when the radius of convergence is finite. (i) (ii) (iv) 2. Given f (x
MATH 223, Linear Algebra Winter, 2008 Assignment 1, due in class January 16, 2008
z 1. Let z = 4 - 2i and w = 8 + i. Find z , w, z + w, z - w, z w and w (all in the form a + bi with a and b real numbers). Find the absolute value of each of these 6
MATH 223, Linear Algebra Winter, 2008 Assignment 7, due in class Wednesday, March 12, 2008 1. The Cayley-Hamilton Theorem says that if A is a square matrix and A (x) is the characteristic polynomial of A, then A (A) = 0. I will not be proving this in
MATH 223, Linear Algebra Winter, 2008 Solutions to Assignment 1
z 1. Let z = 4 - 2i and w = 8 + i. Find z , w, z + w, z - w, z w and w (all in the form a + bi with a and b real numbers). Find the absolute value of each of these 6 numbers. Solut
Calculus III Assignment 4 Due on Friday Oct. 12 1. (a) Find the vector normal to the plane through the points P (1, -1, 0) , Q(2, 1, -1) and R(-1, 1, 2). (b) Find the area of the triangle formed by the above three points. 2. Find the equation of the
Math 223 Homework 9
Due in class April 8
Problem 1: Let A be an n n matrix, and let v be an eigenvector of A with eigenvalue
. Prove that if f (t) is any polynomial, then f (A)v = f ()v.
Solution 1: Note that since v is an eigenvector with eigenvalue , th
Math 223. List of topics.
Here is what we covered so far, with references to the textbook:
(1) Review of complex numbers (1.7) and polynomials (Appendix C)
(2) Vector geometry (1.1 to 1.5) including, e.g., hyperplanes, dot products, and
angles.
(3) Linear
Math 223 Linear Algebra Assignment 3 (Winter 2017)
To be submitted on the 10th floor (in the assignment box) of Burnside Hall
before 4pm on Thu. March 9.
(1) Is the set of functions f R R such that f (x) = 0 for all x < 0 a vector
subspace of the real vec
Math 223 Linear Algebra Assignment 4 (Winter 2016)
To be submitted on the 10th floor of Burnside Hall, in the assignment box,
before 4pm on Tue. Mar. 21.
(1) Let T R2 R3 be the transformation given by T (x, y) = (2x, 3y, 2x + y).
Show that T is linear and
Math 223 Linear Algebra Assignment 2 (Winter 2017)
To be submitted on the 10th floor of Burnside Hall (in the assignment box)
before 4pm on Tuesday Feb. 14th.
Linear systems:
(1) Solve the following system:
x + 2y 3z = 1
2x + 5y + 3z = 4
3x + 8y 13z = 7
[
Math 223 Linear Algebra Assignment 5 (Winter 2017)
To be submitted on the 10th floor of Burnside Hall (in the assignment box)
before 4pm on Tue. Mar. 28.
(1) Let A be a complex n n matrix and let be an eigenvalue of A. Show
that the eigenspace E associate
MATH 223 - LINEAR ALGEBRA
FALL 2016
A SSIGNMENT 1
D UE : S EPTEMBER 20 TH , 8:30
AM
Exercise 1. The cross product operation on vectors in R3 is defined as follows: if
u = (a 1 , a 2 , a 3 ) and v = (b 1 , b 2 , b 3 ), then
u v = (a 2 b 3 a 3 b 2 , a 3 b 1
MATH 223 - LINEAR ALGEBRA
FALL 2016
A SSIGNMENT 3
D UE : O CTOBER 20 TH , 8:30
AM
Exercise 1. Let u, v, and w be three vectors in a linear space V .
(i) True or false: if cfw_ u, v, w is an independent subset, then so is cfw_ u + v, v + w, w + u.
(ii) Tru
MATH 223 - LINEAR ALGEBRA
FALL 2016
A SSIGNMENT 5
D UE : N OVEMBER 15 TH , 8:30
AM
Exercise 1. A symmetric matrix A M n (R) is said to be positive definite if u T Au > 0
for all non-zero vectors u Rn .
(i) Show that a diagonal matrix with positive entries
MATH 223 - LINEAR ALGEBRA
FALL 2016
D EFERRED M IDTERM E XAM
Date: October 27th
Duration: 1 hour and 30 minutes
Note: This sheet has to be returned with the exam booklet.
Last Name & ID :
Exercise 1. True or false? Circle the answer, no justification is n
MATH 223 - LINEAR ALGEBRA
FALL 2016
A SSIGNMENT 2
D UE : O CTOBER 4 TH , 8:30
AM
Exercise 1.
(i) Check that det(AB) = det(A) det(B) for any two 2 2 matrices A and B.
(ii) Assume that a 2 2 matrix satisfies A 10 = 0. Deduce that A 2 = 0.
Exercise 2.
(i) Sh
MATH 223 - LINEAR ALGEBRA
FALL 2016
A SSIGNMENT 6
D UE : N OVEMBER 22 ND, 9:59
AM
Exercise 1. Let V be an inner product space, of dimension n. Let S be an orthonormal basis of V . Show that
u, v = [u]S [v]S
for all u, v V . Recall, [u]S Rn denotes the co
MATH 223 - LINEAR ALGEBRA
FALL 2016
M IDTERM E XAM
Date: October 24th
Duration: 1 hour and 30 minutes
Note: This sheet has to be returned with the exam booklet.
Last Name & ID :
Exercise 1. True or false? Circle the answer, no justification is needed.
T /
The Art of Rice
Spirit and Sustenance in Asia
Roy W Hamilton
W/z'tb contributions by
UCLA Fowler Aurora Ammayao Pattana Kitiarsa
Museum ' Mutua Bahadur Gisele Krauskopff
Of CUltural Hlsmry Francesca Bray Nanditha Krishna
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Math 223 Homework 1
Due in class Jan. 21
Problem 1: Prove the triangle inequality:
u+v u + v
(Hint: Consider the quantity u + v 2 , apply the Cauchy-Schwartz inequality
|u v| u v , and complete the square)
Problem 2: Let u = (2, 1, 1), v = (1, 3, 3), w =
Maths 223 Assignment 10
Calem J Bendell
260467886
[email protected]
April 18, 2014
Problem 1: Do problem 12.9 in the book.
Justify algorithm 12.1, which diagonalises (under congruence) a symmatric matrix A. In this case the algorithm
is in the book, b
Math 223 Homework 8 Solutions
Problem 1: Let V = P3 (t), the space of degree 3 polynomials with real coecients, and
let , be the inner product dened by
1
f, g =
f (t)g(t)dt
1
Apply the Gram-Schmidt process to cfw_1, t, t2 to obtain an orthonormal basis f
Math 223 Homework 8
Due in class April 1
Problem 1: Let V = P2 (t), the space of degree 3 polynomials with real coecients, and
let , be the inner product dened by
1
f, g =
f (t)g(t)dt
1
Apply the Gram-Schmidt process to the basis cfw_1, t, t2 to obtain a
Math 223 Homework 9
Due in class April 8
Problem 1: Let A be an n n matrix, and let v be an eigenvector of A with eigenvalue
. Prove that if f (t) is any polynomial, then f (A)v = f ()v.
Problem 2: Let A be an n n matrix. Using the result of Problem 1, sh
Midterm Vocabulary List
Introduction
The following vocabulary list is designed to help you review for the midterm. With each
of the following terms, check that you understand what the term means and how to use it
in a mathematical sentence. I would sugges
Midterm Exam
October 21st, 2013
Name and Student Number: _
Problem 1 (15 points)
Decide whether the following statements are true or false. If true, you may simply write
"True." If False, provide a counterexample.
(a) Let V be a vector space of dimension
Math 223 Homework 7
Due in class March 25
Problem 1: Let
and
1
2
1
S1 = 2 , 1 , 2
0
1
1
1
1
0
S2 = 1 , 0 , 1
0
1
1
(a) Show that S1 and S2 are bases for R3 .
(b) Let Id : R3 R3 denote the identity map, i.e. the map dened by Id(v) = v for all
v R3
Math 223 Homework 3
Due in class Feb. 3
The goal of the next 7 problems is to give a proof of the following theorem:
Theorem: Let A be an n n matrix that is not invertible. There exists a vector v with
v = 0 such that Av = 0.
Problem 1: Let A be an invert
Math 223 Homework 2
Due in class Sept. 19
Problem 1: Let
2 3 1i
i i
0 , B = i 0 , C =
A= 1 0
0 i
0
0 i
1 1 1
i 2i i
Compute each product in the follow list that is well-dened:
ABC
BAC
CBA
BCA
ACB
CAB
Solution 1:
ABC, BCA and CAB are well-dened, and
MATH 223 - LINEAR ALGEBRA
FALL 2016
Prerequisite: Math 133 - Linear Algebra, or equivalent.
Course description: Review of matrix algebra, determinants and systems of linear equations. Vector spaces, linear operators and their matrix representations, ortho