National University of Singapore
Department of Mathematics
Solution 7 HW4
MA2101 Linear Algebra II
1a. Find the chacteristic polynomial pA (x) of the matrix
3 1 0
A = 0 2 0 .
1 1 2
1b. Find all eigenvalues i of A and eigenspaces Vi (A) as well as the
alge
Tutorial 7 HW4
MA2101 Linear Algebra II
Common Test (covering 1-6; Tut 1-5) be held on Thurs2.10pm3.10pm,19th Mar 2015 at NAK-AUD during the lecture time, at
lecture venue. Students who are absent from the test without an MC will be given
0 mark for the t
Post Exam Remarks
5th May 2015
1. Start every new question with a fresh NEW page, to avoid confusion!
2. Try to do questions (and parts) in order. Worst case: part here, part there.
3. If you are not very confident, spend more time on easy Q, with meaning
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Tutorial 6
1. Let V be an n-dimensional vector space over a eld F and with a
basis B = (v1 , . . . , vn ).
Show that the coordinate map : V Fcn (v [v]B ) is an isomorphism
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Solution 3 HW2
In this Tutorial, you may use the fact from linear algebra 1: n column
vectors are L.I. in Fcn i the matrix A they form has |A| = 0.
1a. Given vectors v1 =
National University of Singapore
Department of Mathematics
Solution 1 HW1
MA2101 Linear Algebra II
1. For the sets below and with operatins given, determine whether or
not they are vector spaces over R. For those that are not, list up all
Axioms in the de
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Solution 10
1. Apply Gram-Schmidt process to transform the basis
cfw_u1 = (1, 1, 1)t , u2 = (0, 1, 1)t , u3 = (0, 0, 1)t
into an orthonormal basis cfw_v1 , v2 , v3 of Fc
Solution to Mid-test MA2101 - Linear Algebra II, 19 March 2015
Question 1 [25 marks]
Let v1 , . . . , v4 be row vectors in the row 3-space V = R3 . Form a matrix
t
t
A := (v1 , . . . , v4 ) M34 (R).
Suppose that A is row-equivalent to the following matrix
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Solution 5 H3
Note. In the solutions below, we often use Equivalent linear transformation denition theorem in 6.
1. Determine whether or not the maps below are linear tran
National University of Singapore
Tutorial 5 H3
MA2101 Linear Algebra II
1. Determine whether or not the maps below are linear transformations. If yes, prove it; if no, give a counter-example.
[Suggested Answers: Y, N, Y, N, N, N.]
T1 : R[x] R[x]
f (x) = a
National University of Singapore
Department of Mathematics
Solution 9 HW5
MA2101 Linear Algebra II
1. Let A M4 (F ) and let J Mn (F ) be a Jordan canonical form of
A (you may assume A = J to do the questions below).
Find J and the characteristic and minim
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Solution 2
1. Let S, T be subsets of a vector space V over a eld F . Show that
Span(S T ) = Span(S) + Span(T ).
Solution for Q1:
To show
Span(S T ) Span(S) + Span(T )
take
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Solution 4
1a. Find a subset of row vectors v1 , ., v5 below, that becomes a basis
of Spancfw_v1 , ., v5 ;
express each vector not in the basis as a linear combination of
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Solution 6
1. Let V be an n-dimensional vector space over a eld F and with a
basis B = (v1 , . . . , vn ).
Show that the coordinate map : V Fcn (v [v]B ) is an isomorphism
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Tutorial 10
1. Apply Gram-Schmidt process to transform the basis
cfw_u1 = (1, 1, 1)t , u2 = (0, 1, 1)t , u3 = (0, 0, 1)t
into an orthonormal basis cfw_v1 , v2 , v3 of Fc
National University of Singapore
Department of Mathematics
Tutorial 9 HW5
MA2101 Linear Algebra II
1. Let A M4 (F ) and let J Mn (F ) be a Jordan canonical form of A
(you may assume A = J to do the questions below).
Find J and the characteristic and minim
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
1. Consider the matrix
A1 0
0 A2
A= .
.
.
.
.
.
.
.
.
0 0
0
0
.
.
.
0
0
.
.
.
Solution 8
0 Ar
where Ai Mni (F ) are square matrices.
Show that the minimal polynomial
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ANGLO-CHINESE JUNIOR COLLEGE
MATHEMATICS DEPARTMENT
MATHEMATICS
Higher 2
Paper 1
9740
14 August 2008
JC 2 PRELIMINARY EXAMINATION
Time allowed: 3 hours
Additional Materials: List of Formulae (MF15)
READ THESE INSTRUCTIONS FIR
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SAINT ANDREWS JUNIOR COLLEGE
PRELIMINARY EXAMINATION
MATHEMATICS
Higher 2
Paper 1
Thursday
9740/01
30 August 2007
3 hours
Additional materials : Answer paper
List of Formulae(MF15)
Cover Sheet
READ THESE INSTRUCTIONS FIRST
Wr
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RAFFLES JUNIOR COLLEGE
JC2 Preliminary Examination 2007
9740/01
MATHEMATICS
Higher 2
Paper 1
12 September 2007
3 hours
Additional materials : Answer Paper
List of Formulae (MF15)
READ THESE INSTRUCTIONS FIRST
Write your name
www.teachmejcmath-sg.webs.com
2
IJC Prelim 2007 P1
1
The curve with equation y = ax2 + bx + c passes through the points (1, 7), (3, 11) and
(5, 99). Find the equation of the curve.
[4]
dy
0.4 y (10 y ) and y 5 when x 1 , find an expression for y in
dx
te
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Anderson Junior College
Preliminary Examination 2007
H2 Mathematics Paper 1
1
4
in ascending powers of x up to and including the term in x2, stating the
3
2 x
range of x for which the expansion is valid. Find the coefficient
Studying Strategies
General Paper
Read newspapers, Time magazines, Readers Digest, National Geographic
Watch news, BBC, CNA, CNN
Read sample essay
Refer to dictionary everything when you discover new words (Write them down in an exercise
book)
Practi
National University of Singapore
Department of Mathematics
Tutorial 9 HW5
MA2101 Linear Algebra II
1. Let A M4 (F ) and let J Mn (F ) be a Jordan canonical form of A
(you may assume A = J to do the questions below).
Find J and the characteristic and minim
Tutorial 7 HW4
MA2101 Linear Algebra II
1a. Find the chacteristic polynomial pA (x) of the matrix
3 1 0
A = 0 2 0 .
1 1 2
1b. Find all eigenvalues i of A and eigenspaces Vi (A) as well as the
algebraic and geometric multiplicities of i .
[Suggested Answer
National University of Singapore
MA2101 Linear Algebra II
Tutorial 2
1. Let S, T be subsets of a vector space V over a eld F . Show that
Span(S T ) = Span(S) + Span(T ).
2a. Consider the line L1 = cfw_t(1, 2, 3) | t R and the plane
P1 = cfw_(x, y, z) R3 |
National University of Singapore
Tutorial 5 H3
MA2101 Linear Algebra II
1. Determine whether or not the maps below are linear transformations. If yes, prove it; if no, give a counter-example.
[Suggested Answers: Y, N, Y, N, N, N.]
T1 : R[x] R[x]
f (x) = a
1
2
When 1.00 dm3 of aqueous [Ni(NH3)6]2+ solution was added to an equal volume of sodium cyanide
solution, colour changes are observed.
Given that 0.6 moles of [Ni(NH3)6]2+, 1.0 moles of CN- and 0.6 moles of [Ni(CN)4]2- were mixed together,
and when equi
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Tutorial 6
1. Let V be an n-dimensional vector space over a field F and with a
basis B = (v1 , . . . , vn ).
Show that the coordinate map : V Fcn (v 7 [v]B ) is an isomorp
National University of Singapore
Department of Mathematics
Tutorial 9 HW5
MA2101 Linear Algebra II
1. Let A M4 (F ) and let J Mn (F ) be a Jordan canonical form of A
(you may assume A = J to do the questions below).
Find J and the characteristic and minim
National University of Singapore
Department of Mathematics
Tutorial 3 HW2
MA2101 Linear Algebra II
In this Tutorial, you may use the fact from linear algebra 1: n column
6 0.
vectors are L.I. in Fcn iff the matrix A they form has |A| =
1a. Given vectors v
National University of Singapore
Department of Mathematics
Tutorial 1 HW1
MA2101 Linear Algebra II
1. For the sets below and with operatins given, determine whether or
not they are vector spaces over R. For those that are not, list up all
Axioms in the de