September 28, 2015
1
TEST 1 SOLUTIONSMATH 2107
1. Indicate whether each of the ve statements below is TRUE or FALSE.
Provide some justication for each decision.
/2
/2
/2
(a) The vector space R2 is innite-dimensional.
Solution: False. The standard basis fo
MATH 2107 - Solutions to Assignment 1
1 [8 points]. Yes, H is a subspace of Mnn (R). Indeed, H is a subset of Mnn (R);
the zero n n matrix 0nn H because 0T = 0nn ; (A + B)T = AT + B T = A + B
nn
for any A, B H implies A + B H; and (cA)T = cAT = cA for any
MATH2107
Midterm #2
Answers
1. The transition matrix P7E is just the matrix of coordinate vectors for the vs in terms of
the standard basis vector. Using these coordinate vectors, we solve
so that v = (5a+2b-2c-4d)v1 + (-2a-b+c+2d)v2 - dv3 + (-4a-2b+c+d)v
Linear Algebra II
MATH 2107 B
Course outline Winter 2016
School of Mathematics and Statistics
Carleton University
Instructor: Ranjeeta Mallick 5250 Herzberg, 613-520-2600 ext. 1983
E-mail:
[email protected]
Textbook: Linear Algebra and Its Applica
MATH 2107
Tutorial 1
Q1. Is the following set of vectors a vector space with the
indicated vector addition and scalar multiplication? Explain.
a) The set 2 with usual additions and scalar multipli cations
x cx
defined as c
y cy
Ans: The given set is
CARLETON UNIVERSITY
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2107 Linear Algebra II
Solutions to Test 2
1.
(a) To show that B1 is a basis for V = P2 , we first check for linear independence. Consider the linear combination
a 1 + b(1 + x) + c(1 + x + x2 )
MATH 2107A
TEST 4
NOVEMBER 20, 2009
This test has two parts with a total of 30 marks. The test cannot be taken out from the examination room. Only nonprogrammable calculators are allowed. Show all your work. Duration: 50 minutes. NAME (in ink):
3 1 3 2 an
A. Alaca
Linear Algebra II
Winter 2017
LINEAR ALGEBRA II
LECTURE NOTES
c
Ay
se Alaca
Last modified: January 23, 2017
(These Lecture Notes replace neither the Text Book nor the Lectures)
RANK
Section 4.6
1
A. Alaca
Linear Algebra II
Winter 2017
RANK
Defini
A. Alaca
Linear Algebra II
Vector Spaces
MATH 2107 LINEAR ALGEBRA II
LECTURE NOTES
c
Ay
se Alaca
Last modified: January 5, 2017
(These Lecture Notes replace neither the Text Book nor the Lectures)
Section 4.1
VECTOR SPACES and SUBSPACES
These notes may be
A. Alaca
Linear Algebra II
1
MATH 2107 LINEAR ALGEBRA II
LECTURE NOTES
c
Ay
se Alaca
Last modified: January 10, 2017
(These Lecture Notes replace neither the Text Book nor the Lectures)
Section 4.2
NULL SPACES, COLUMN SPACES, and LINEAR TRANSFORMATIONS
Th
WINTER 2017
MATH 2107B, Linear Algebra II
Instructor: Ayse Alaca,
Herzberg Physics, Office #4376
Tel: (613) 520 2600 (Ext. 2133)
http:/www.math.carleton.ca/~aalaca/
Textbook: Linear Algebra and Its Applications, 5E, by David C. Lay, Steven R. Lay, Judi J.
A. Alaca
MATH 2107 Linear Algebra II
LECTURE NOTES
c
Ay
se Alaca
Last modified: January 17, 2017
(These Lecture Notes replace neither the Text Book nor the Lectures)
DIMENSION of a VECTOR SPACE
Sections 4.3, 4.4 and 4.5
These lecture notes may be incomple
Linear Algebra II
Quadratic Forms
December 7, 2016
Quadratic Forms ()
Linear Algebra II
December 7, 2016
1/5
Definition
Definition
An expression of the form
F (x, y ) = ax 2 + by 2 + cxy
is called a quadratic form in x and y .
Similarly,
F (x, y , z) = ax
MATH2107
SAMPLE TEST #1
ANSWERS
1. (a) The zero function is not in U (since it cannot take on the value 1) so U is not a
subspace. (Its not closed under addition nor scalar multiples, either!)
(b) The zero function is certainly continuous, so W is not emp
MATH2107
Test #3
1. We have <u,v> = 0 and <u,u + v> = 0 so that
0 = <u,u> + <u,v> = <u,u> + 0 = <u,u>.
Then <u,u> = 0 gives u = 0, as required.
2. Each row vector has length 2 so our orthonormal basis is
= cfw_ [ ], [ - - ], [ - - ] .
Then
v.[ ] = (1 + 5
MATH 2107
Tutorial 3
Feb 12 2015
Please work in teams of 4. At the end of the tutorial every team hands in one set of
solutions with everybodys name and student number PRINTED.
Your TA is here to help you- dont be shy to ask questions!
*
Name _
Student #_
MATH 2107
Tutorial 2
Feb 5 2015
Please work in teams of 4. At the end of the tutorial every team hands in one set of
solutions with everybodys name and student number PRINTED.
Your TA is here to help you- dont be shy to ask questions!
*
Name _
Student #_
MATH 15a: Linear Algebra
Exam 2, Solutions
1. Let A be the matrix
2
4
0
3 4 2
6 13 1
0
2 2
(a) (4 points) If A is the matrix for a linear transformation T : Rn Rm , what are
m and n?
Answer: m = 4, n = 3.
(b) (8 points) Find a basis for im A.
Answer: Row
Stud Guidelines for POD 30-35 ts: CHEMICAL BONDING
Tues Dec 13 blue d3 IWGd Dec 14 red d3 - Close out ear2016 onacom eiiin note.
N.B. As of HW due Mon, Dec 12 (anchor), ALL sections of Ch 9: Covalent Bondin- will have been HW reading.
Emphasis on delta EN
MATH2107
Tutorial #3
Fri10thOct.2014
1. Let A and B be nn matrices with eigenvalues , respectively. Suppose that there is
a non-zero vector v in n which is an eigenvector for A with respect to and also an
eigenvector for B with respect to . Verify that is
MATH2107
Tutorial #3
Solutions
1. Since the vector v is an eigenvector for A with respect to and also an eigenvector
for B with respect to , we have
Av = v
and
Bv = v
so that
ABv = A(v) = Av =(v) = v
which shows that is an eigenvalue for AB with v as an e
MATH 2107A Final/Deferred Exam
December 2008
Page
PART I: Multiple Choice Questions. Three marks each.
No partial mark. Circle the correct answer.
1. Let A be a 5 7 matrix such that dim null(A) = 3. What is rank(A) ?
(a) 2
2. Let B =
(b) 3
1 0
0 0
,
(c) 4
CARLETON UNIVERSITY
SCHOOL OF MATHEMATICS AND STATISTICS
MATH 2107
Linear Algebra II
FALL 2016
Preliminary COURSE OUTLINE
Objectives of the course: this is a second course in the Linear Algebra sequence.
Its main content is an introduction to more advance
LINEAR ALGEBRA II. LECTURE NOTES
CHAPTER 2. LINEAR TRANSFORMATIONS
INNA BUMAGIN
Contents
1. Definition and Examples
1.1. Properties of Linear Transformations
2. The Kernel and Image of Linear Transformation
2.1. One-to-one and Onto Transformations
2.2. Th
Complex Numbers. Review
Contents
1. Definition
2. Operations with complex numbers
3. Properties of Operations with Complex Numbers
4. Geometric Interpretation of C
4.1. The Polar Form of a Complex Number
5. Multiplication of Complex Numbers
6. De Moivres
LINEAR ALGEBRA II. LECTURE NOTES. FALL 2016
CHAPTER I. REVIEW
INNA BUMAGIN
Contents
1. The Invertible Matrix Theorem
2. Vector Spaces
2.1. Linear combinations of vectors and span
3. Subspaces
3.1. Lines and planes in R3
3.2. The Null space and the Image s