Mid-term Exam #1
MATH 205, Fall 2014
Name:
Instructions: Please answer as many of the following questions as possible. Show
all of your work and give complete explanations when requested. Write your nal
answer clearly. No calculators or cell phones are al

MATH 205A,B - LINEAR ALGEBRA
WINTER 2013
QUIZ 5
NAME:
Section:(Circle one)
A(1 : 10)
B(2 : 40)
Show ALL your work CAREFULLY.
abc
adg
(a) Suppose det d e f = 3. Find det 2b 2e 2h.
ghi
cf
i
T
adg
adg
adg
abc
det 2b 2e 2h = 2 det b e h = 2 det b e h = 2 det

8am
~i
=
1. Suppose A ~
[
~
000
,
NaJlle~J'frJ sd.a
03/06/099 -18 Quiz 05 page I
12 3
Math 205B&C
1:10 pm
4]
1
and A is some 4 x 4 matrix.
Suppose that A rv Al rv A2 rv A3 rv A4 = A~, and the following row operations are applied sequentially
to turn A int

MATH240: Linear Algebra
Quiz #6 solutions
6/22/2015 Page 1
Write legibly and show all work. No partial credit can be given for an unjustied, incorrect
answer. Put your name in the top right corner.
3 3 1
1. (Six points) Find the LU -factorization of the m

H2 Maths Paper 1 Practice Session 2
4 August 2012
Time: 9 am 12 pm
Note: You are advised to spend the first 5 minutes to look through the questions and do short
annotations before deciding on which question to begin. Answer all questions.
1
Find the real

H2 Maths Paper 1 Practice Session 1_Solutions
1.
Objective: To test awareness of inequality with repeated roots
9
9
4 4
0
2
2 x x
2 x x2
9
4+ 2
0
x + x2
4 x2 + 4 x + 1
2
0
x + x2
2
( 2 x + 1) 0
( x + 2 )( x 1)
Using sign test or graphical method, x < 2

H2 Mathematics 9740 Paper 1 Solutions for Practice Session 2
Question 1
5 x 2 3x + 1
ax + 1
b
= 2
+
2
( x + 1)( x 2) ( x + 1) ( x 2)
5 x 2 3x + 1 = (ax + 1)( x 2) + b( x 2 + 1)
Comparing constant term:
1 = 2 + b b = 3
2
Comparing coefficients of x : 5 =

NANYANG JUNIOR COLLEGE
JC 2 MID-YEAR EXAMINATION
Higher 2
CANDIDATE
NAME
TUTOR
NAME
CLASS
PHYSICS
9646/01
Paper 1 Multiple Choice
5 July 2011
1 hour 15 minutes
Additional Materials:
Multiple Choice Answer Sheet
READ THESE INSTRUCTIONS FIRST
Write in soft

1
NANYANG JUNIOR COLLEGE
JC 2 MID-YEAR EXAMINATION
Higher 2
CANDIDATE
NAME
TUTOR
NAME
CLASS
PHYSICS
9646/01
5 July 2011
Paper 1 Multiple Choice
1 hour 15 minutes
Additional Materials:
Multiple Choice Answer Sheet
READ THESE INSTRUCTIONS FIRST
Write in sof

NANYANG JUNIOR COLLEGE
JC 2 MID YEAR EXAMINATION
Higher 2
CANDIDATE
NAME
TUTORS
NAME
CLASS
PHYSICS
9646/02
Paper 2 Structured Questions
5 July 2011
1 hour 45 minutes
Candidates answer on the Question Paper.
No Additional Materials are required
READ THESE

For
Examiners
Use
Mid year H2 P2 Physics Solution
1
A body X of mass m is projected towards a stationary body Y of mass M with a velocity of
u. The bodies eventually collide and the velocity of X after the collision is v.
u
v
Y
X
before collision
(a)
X
Y

NANYANG JUNIOR COLLEGE
JC 2 MID YEAR EXAMINATION
Higher 2
CANDIDATE
NAME
TUTORS
NAME
CLASS
PHYSICS
9646/03
Paper 3 Longer Structured questions
29 June 2011
2 hours
Candidates answer on the Question Paper.
No Additional Materials are required
READ THESE IN

NANYANG JUNIOR COLLEGE
JC 2 MID YEAR EXAMINATION
Higher 2
CANDIDATE
NAME
TUTORS
NAME
CLASS
PHYSICS
9646/03
Paper 3 Longer Structured questions
29 June 2011
2 hours
Candidates answer on the Question Paper.
No Additional Materials are required
READ THESE IN

1
BLOCK TEST PRACTICE PAPER A
1
A sequence u1 , u2 , u3 , is such that u1 = 0 and (1 + n)u n +1 = n + u n for n Z + .
Prove by mathematical induction that 3un +1 = 3
(i)
(i) Let Pn be the statement: 3un +1 = 3
3
for n Z + .
( n + 1)!
3
for n Z + .
( n +

1
1.
A sequence u1 , u2 , u3 , is such that u1 = 0 and (1 + n)u n +1 = n + u n for n Z + .
3
(i)
Prove by mathematical induction that 3un +1 = 3
for n Z + .
( n + 1)!
(ii)
State the limit of the sequence.
Solution:
Let Pn be the statement: 3un +1 = 3
3

n
1
(a)
Prove by mathematical induction, that
4r
1
2
r =1
1
=
n
for all positive integers, n.
2n + 1
[4]
(b)
Hence, or otherwise, find the sum to infinity of the series
1 1 1
+ + + .
15 35 63
1
(a)
[2]
2011 SRJC MYE P1Q7
n
1
Let Pn be the statement
=
2

MH1201 - LINEAR ALGEBRA II
AY 2014-2015
Dr. Le Hai Khoi ([email protected])
Division of Mathematical Sciences, SPMS, NTU
2
Preface
These lecture notes are to be used together with the textbook [1] and the reference-book
[2]. During the lectures we will fo

PRACTICING PROOFS
This le contains two sets of problems to practice your ability with proofs. Solutions to the rst set of problems are provided. The solutions to the second set of
problems are intentionally left to the reader (as an incentive to practice!

MA1101R
Linear Algebra I
AY 2010/2011 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
with credits to Luo Xuan and Zhang Manning
MA1101R Linear Algebra I
AY 2010/2011 Sem 1
Question 1
(a) Proof:
First we show that span

MT210 MIDTERM 1 SAMPLE 1
ILKER S. YUCE
FEBRUARY 16, 2011
QUESTION 1. SYSTEMS OF LINEAR EQUATIONS
Determine the values of k such that the linear system
9x1
kx1
+ kx2
+
x2
=
9
= 3
is consistent.
ANSWER
We apply row-reduction algorithm to the augmented matri

MT210 MIDTERM 1 SAMPLE 2
ILKER S. YUCE
FEBRUARY 16, 2011
QUESTION 1. SYSTEMS OF LINEAR EQUATIONS
The augmented matrix of a linear system has the form
[
]
a
1 1
2 a1 1
Determine the values of a for which the linear system is consistent.
1
QUESTION 2. ROW R

MT210 MIDTERM 1 SAMPLE 2
ILKER S. YUCE
FEBRUARY 19, 2011
QUESTION 1. SYSTEMS OF LINEAR EQUATIONS
The augmented matrix of a linear system has the form
[
]
a
1 1
2 a1 1
Determine the values of a for which the linear system is consistent.
ANSWER
We apply row

MT210 MIDTERM 1 SAMPLE 3
ILKER S. YUCE
FEBRUARY 16, 2011
QUESTION 1. SYSTEMS OF LINEAR EQUATIONS
Verify that if ad bc = 0, then the system of equations
ax1
cx1
+ bx2
+ dx2
has a unique solution.
1
= r
= s
QUESTION 2. ROW REDUCTION AND ECHELON FORMS
Write

MT210 MIDTERM 1 SAMPLE 3
ILKER S. YUCE
FEBRUARY 20, 2011
QUESTION 1. SYSTEMS OF LINEAR EQUATIONS
Verify that if ad bc = 0, then the system of equations
ax1
cx1
+ bx2
+ dx2
= r
= s
has a unique solution.
ANSWER
We apply row-reduction algorithm to the augme

MT210 MIDTERM 1 SAMPLE 4
LKER S. YCE
FEBRUARY 16, 2011
Surname, Name:
QUESTION 1. SYSTEMS OF LINEAR EQUATIONS
Solve the linear system
x1
x1
2x1
+ x2
x2
x2
+
x3
+
x3
+ 2x3
1
=
4
= 2
=
2
QUESTION 2. ROW REDUCTION AND ECHELON FORMS
Find the row reduced ech

TEST 1
Math 205 H
mmnl Name: 415% L/
bg- LngmymnlmlfhythLMmﬁnM
Read all of the fullﬂwing inﬁjrmatinu before starting the m:
I Show all wk, clearly and in under if you want to get full Bandit (matrices can be reduced into
REEF with calculator without 513mm

MT210 MIDTERM 1 SAMPLE 4
LKER S. YCE
FEBRUARY 16, 2011
Surname, Name:
QUESTION 1. SYSTEMS OF LINEAR EQUATIONS
Determine when the augmented matrix represents a
1
0 2
2
1 5
1 1 1
consistent linear system.
a
b
c
ANSWER
We need to reduce the given matrix:
1

MATH 54 QUIZ 3 SOLUTIONS
PEYAM RYAN TABRIZIAN
1. (4 points) Find the inverse of
invertible)
0 1
0 3
1 1
1 1
Form the big matrix:
0
0
A I =
1
1
1
3
1
1
And row-reduce until you
identity:
1 0
0 1
I A1 =
0 0
0 0
the following matrix (or say its not
0 0
2 1

NANYANG TECHNOLOGICAL UNIVERSITY
SEMESTER I EXAMINATION 2012-2013
MH1200/MTH 114 Linear Algebra I
November 2012
TIME ALLOWED: 2 HOURS
INSTRUCTIONS TO CANDIDATES
1. This examination paper contains FOUR (4) questions and comprises
FIVE (5) printed pages.
2.

MH1201
Chapter 2: Linear Transformations
MH1201
1 / 153
Outline
1
Basic Notations
2
The Matrix Representation of a Linear Transformation
3
Composition of Linear Transformations
4
Invertibility and Isomorphisms
5
The Change of Coordinate Matrix
MH1201
2 /