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
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
Bonus Homework
To be turned in any time before 5:00 PM April 14th
Problem 1: Do problem 12.9 in the book.
Problem 2 : Let
1 6 2
A= 6 6 0
2 0 3
Find a matrix P such that P T AP is diagonal.
What is the signature of A?
Problem 3 : Let V denote the space
Midterm Vocabulary List
April 9, 2014
Introduction
The following vocabulary list is designed to help you review for the nal exam.
The sections of the book that we have covered in class since the midterm are the following:
Chapter 6 section 5
Chapter 7 s
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
Maths 223 Assignment 10
Calem J Bendell
260467886
calembendell@live.com
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
Maths 223 Assignment 10
Calem J Bendell
260467886
calembendell@live.com
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
McGill University
Department of Mathematics and Statistics
MATH 223 Linear Algebra
Winter 2014
Instructor: Dr. William Cavendish
Oce: 1212 Burnside Hall
Email: william.cavendish@mcgill.ca
Oce hours: Wednesday 8:30-11:30
Textbook: Schaums Outline of Linear
McGill University
Department of Mathematics and Statistics
Linear Algebra /MATH 223
Midterm examination
February 16, 2009
General instructions:
1. This is a CLOSED BOOK exam. Resources such as books, notes, computers, black
berries, etc. that could be use
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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 Homework 4
In Problem 1, let P2 (t) denote the space of degree 2 polynomials with real coecients.
Problem 1: Let f (t) = 2t2 + t, g(t) = t2 t + 1, and h(t) = t 3. Do f, g and h span
P2 (t)? If so, write t2 + t + 1 as a linear combination of f , g
The Proof Bank
This document contains a list of theorems whose proofs are of a similar level of diculty
to those that will appear on the nal. Like on the midterm, there are two questions on the
nal (on the nal they will be worth 15/100 points each) where
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 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