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 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
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
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
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 #_
Linear Algebra
7. Symmetric Matrices and Quadratic Forms
CSIE NCU
1
7. Symmetric Matrices and Quadratic Forms
7.1 Diagonalization of symmetric matrices 2
7.2 Quadratic forms . 9
7.4 The singular value decomposition . 21
Linear Algebra
7. Symmetric Matrice
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 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
basis . Let T : P2 ; T (a + bx + cx ) = (a + b, c) be a li
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 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 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 you- dont be shy to ask questions!
*
Name _
Student #_
MATH 2107 - Assignment 1
Due June 1 at the beginning of tutorial
Recall that Mmn (R) the set of m n matrices over R with the usual operations
of matrix addition and matrix scalar multiplication is a vector space over R. Let
A Mmn (R). We denote the determ
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
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
MATH2107
Fri19th Sep.2014
TUTORIAL #1
1. Subtracting gives
av - bv = 0
and so (a - b)v = 0.
Hence, using the result from class, either a - b = 0 or v = 0. Thus a = b or v = 0.
2. (i) W is not empty since O22 is in it. For A =
A+B=
which is also in W. kA =
MATH2107
TUTORIAL #1
Fri19th Sep.2014
1. Suppose that V is a vector space and
av = bv
where a, b 0 and v 0 V. Show that either a = b or v = 0.
2. For each vector space V and given subset W, determine whether or not W is a
subspace of V.
(i) V = M22 (space
MATH2107
Tutorial #2
Answers
1. We try to solve AX = v so we set up the matrix [A : v].
and we can see that one solution will be x4 = 2, x3 = 0, x2 = 1, x1 = 2. (There are other
solutions where x3 0.)
Hence v 0 col(A) and we have v = 2c1 + c2 + 2c4.
2. If
MATH2107
1. Let A be the matrix
Fri26thSep.2014
Tutorial #2
. Is v =
in col(A)?
If so, write v as a linear combination of the columns of A.
2. Let f (x) = sin2x, g(x) = cos2x, h(x) = sin2x be vectors in the space of all real functions
and take V as the su
MATH2107
TEST #1
Solutions
1. The columns of A and B satisfy the same relations and columns 1, 2, 3 and 5 of B are
linearly independent. Since the other columns are linear combinations, we get our basis
for V as
cfw_ c1 , c 2 , c 3 , c 5
(when A = [c1, c
Disclaimer
These sample questions may or may not be an indication as to the kind of problems that you may
see in the future. Any passing resemblance to some future test may be entirely coincidental.
Users proceed at their own risk and any anguish, physica
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
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
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