MATH 2107B
TEST 1
February 4, 2016
Only nonprogrammable calculators are allowed.
Duration: 50 minutes.
Total marks: 30
NAME (in ink):
STUDENT NO (in ink):
PART I: [6] True/False questions. Circle the
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 =
CARLETON UNIVERSITY
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2107 Linear Algebra II
Test 2
6th July 2009
Answer all the questions in the booklet provided
Time: 1 hour 30 minutes
1. Let V = P2 , and le
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 inn
CARLETON UNIVERSITY
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2107 Linear Algebra II
Test 1
8 June 2009
Answer all the questions in the booklet provided
Time: 1 hour 30 minutes
In all questions, you ne
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]
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
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 a
CARLETON UNIVERSITY
SCHOOL OF MATHEMATICS AND STATISTICS
MATH2107 Linear Algebra II
Solutions to Test 1
1.
(a)
1
1
2
u + v = 0 + 2 = 2 .
1
3
4
(1 mark)
(b)
1
3
3u = 3 0 = 0 .
1
3
(1 mark)
(c)
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 cat
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 you
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
Solutions to Problem Set 2
2. (a) Since A~x = ~0, ~x N ul(A). To find a spanning set of N ul(A), we apply
row reductions:
i
2i
5
i
0
0
0
0
5
5
5i
1
0
3
3i
A ~0 =
1 2 + 5i
4
1 2 + 5i
4
0 1 1i 0
0
0
Solutions to Problem Set 4
1 0
0 1
0 0
2. In Problem 3 from PS3, the set
,
,
is a basis for
0 0
1 0
0 1
H; and so, dim(H) = 3. Note that dim(H) = 3 < 4 = 2 2 = dim(M22 (R).
3. Since
1 0
0 1
0 0
=a
MATH 2107
Tutorial 5
Mar 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
Linear Algebra II
MATH 2107 - Fall 2017
Instructor: Ming Ming Zhang
Office: 4348 HP (Herzberg Laboratories)
Email: [email protected]
Office hour: Thursday 11:30-13:00
1
Introduction
Linear Al
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
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
MATH 2107
Tutorial 4
Mar 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
MATH 2107B
TEST 3
March 19, 2015
Duration: 50 minutes
Total marks:30
NAME :
STUDENT NO:
1 1
[6] 1. Let B = cfw_x + 1, x 2 ,3 and B1 = cfw_1, x, x 2 be two bases of P2 and D = , be a
1 1
2
2
2
bas
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
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
Solutions to Problem Set 1
1 0
0 0
2. (a) No, H is not a subspace of M22 (R). Consider two matrices A =
0 0
and B =
. Then det(A) = det(B) = 0; and so, A, B H. However,
0 1
1 0
A+B =
and det(A + B) =
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 q
LINEAR ALGEBRA II. LECTURE NOTES
CHAPTER 4. ORTHOGONALITY
INNA BUMAGIN
Contents
1. Orthogonality
1.1. Unit Vectors
1.2. Orthogonal Vectors
1.3. Orthogonal and Orthonormal Sets of Vectors
1.4. The coor
LINEAR ALGEBRA II. LECTURE NOTES
CHAPTER 3. DIAGONALIZATION
INNA BUMAGIN
Contents
1. Diagonalization
1.1. Definitions of Eigenvalues and Eigenvectors
1.2. The Characteristic Polynomial of a matrix A
1
1. Determine whether or not the following matrices are diagonalizable. Diagonalize those
that are diagonalizable.
2 3
(a) A =
4 1
2 1
(b) A =
1 4
8 4 4
8 4
(c) A = 4
4
4 8
7 9
8
2. Calculate A where
Redox practice worksheet
Name:
1.
In which substance is the oxidation number of
nitrogen zero?
A.
2.
B.
N2
C.
NO2
D. N2 O
+6
B.
+2
C.
4
D.
Al0
B.
Cr3+
C.
Al3+
In the reaction 2K + Cl2 ! 2KCl, the spec