MA1101R
Linear Algebra I
AY 2009/2010 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
with credits to Chang Hai Bin
MA1101R Linear Algebra I
AY 2009/2010 Sem 1
Question 1
(a)
(i) Assume the number of orange, grapefruit
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 8
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 10
(19/10 - 23/10).
You are advised to revise Sections 5.
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 9
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 11
(26/10 - 30/10).
You are advised to revise Sections 5.
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 10
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 12
(2/11 - 6/11).
You are advised to revise Chapter 6 of
Solutions to Tutorial 10
1. (a) By the result of Question 4(a) of Tutorial 9, the matrix is diagonalizable.
3 1
2 0
1
Let P =
. Then P AP =
.
1 1
0 2
(b) By the result of Question 4(b) of Tutorial 9, the matrix is not diagonalizable.
(We can only find thr
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 11
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 13
(9/11 - 13/11).
You are advised to revise Chapter 7 o
Solutions to Tutorial 7
1. (a) Since (0, 1, 1) and (1, 2, 0) satisfy the equation 2x y + z = 0, S is a subset
of V .
S is linearly independent because the two vectors are not scalar multiples of
each other. (See Example 3.4.3.4.)
As dim(V ) = 2, by Theore
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 7
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 9 (12/10
- 16/10).
You are advised to revise Sections 3.7
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 6
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 8 (5/10
- 9/10).
You are advised to revise Sections 3.4 t
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 5
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 7 (28/9
- 2/10).
You are advised to revise Sections 3.1 t
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 4
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 6 (14/9
- 18/9).
You are advised to revise Sections 2.5 t
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 3
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 5 (7/9
- 11/9).
You are advised to revise Sections 2.3 to
Solutions to Tutorial 11
1. (a) T1 is a linear transformation.
x
x
1 0 x
T1
=
=
y
2x
2 0 y
x
for
R2 .
y
1 0
The standard matrix is
.
2 0
(b) T2 is not a linear transformation.
(c) T3 is a linear transformation.
x
0 0 0
0
x
T3 y = y x = 1 1 0
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2015/2016
MA1101R Linear Algebra
Homework Assignment 2
Please write your name, matriculation card number and tutorial group number
on the answer script, and submit during either Group 1s lecture on 28th Septemb
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2015/2016
MA1101R Linear Algebra
Homework Assignment 3
Please write your name, matriculation card number and tutorial group number
on the answer script, and submit during either Group 1s lecture on 26th October
Answers/Solutions of Exercise 6
(Version: November 6, 2012)
1. (a) The characteristic equation is ( + 1)( 3) = 0; eigenvalues are 1 and
3; cfw_(0, 1)T is a basis for E1 and cfw_(1, 2)T is a basis for E3 .
(b) The characteristic equation is ( 2)2 = 0; th
Answers/Solutions of Exercise 6
(Version: November 2, 2012)
1. (a) The characteristic equation is ( + 1)( 3) = 0; eigenvalues are 1 and
3; cfw_(0, 1)T is a basis for E1 and cfw_(1, 2)T is a basis for E3 .
(b) The characteristic equation is ( 2)2 = 0; th
Answers/Solutions of Exercise 4
(Version: October 16, 2012)
Remark: Please note that bases for vector spaces are not unique. In the following,
if a question asks for a basis, the answer given is only one of the possible answers.
1. In order to answer (iv)
Answers/Solutions of Exercise 3
(Version: September 21, 2012)
1. u = (1, 3), v = ( 3, 1), u + v = (1 3, 1 + 3),
3u 2v = (3 + 2 3, 2 + 3 3).
2. (a) Substituting (x, y) = (1, 2) and (2, 1) into the equation ax + by = c, we
has a system of linear equations
a
MA1101R
Linear Algebra 1
AY 2008/2009 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
with credits to Associate Professor Victor Tan
solutions prepared by Koh Chuen Hoa, Terry Lau Shue Chien
MA1101R Linear Algebra 1
AY
Chapter 6
Diagonalization
In this chapter, all vectors are written as column vectors.
Chapter 6 Diagonalization
Section 6.1
Eigenvalues and Eigenvectors
An example
(Example 6.1.1)
Suppose for each year, 4% of the rural population moves
to the urban distri
Chapter 5
Orthogonality
Chapter 5 Orthogonality
Section 5.1
The Dot Product
Lengths of vectors in
2 (Discussion 5.1.1.2)
Let u = (u1, u2) be a vector in
2.
Then the length of u is given by
|u| = u12 + u22 .
y
u = (u1, u2)
|u|
u2
(0, 0)
u1
x
Lengths of vec
MA1101R Homework I
Common mistakes
September 26, 2015
1
Questions 1-3
Main minor (comparable to others) mistake here: when you doing row operations, always check, that you
are not dividing by zero, like row transformation R3 1 R2 in Q2 without considering
0
1
1. Let A =
1
0
0
1
1
0
1
0
1
1
1
0
2
3
1
0
3
5
0
0
.
0
0
(i) Use the Gauss-Jordan Elimination to reduce A to the reduced row-echelon
form. (Indicate the elementary row operation used in each step.)
(ii) Let T : R6 R4 be a linear transformation such
LESSON 1: MATRIX OPERATIONS AND SOLVING LINEAR SYSTEMS
Abstract. In this laboratory session, we introduce some very basic MATLAB commands
for performing matrix operations and solving linear systems.
1. Introduction
1.1. Accessing MATLAB.
In this course, w
Solutions to Tutorial 2
1. (a) No. The two equations represent two non-parallel lines in the xy-plane and
the lines intersect at the origin. So the system has only the trivial solution.
(b) No. It is obvious that the linear system has only the trivial sol
Solutions to Tutorial 2
1. (a) No. The two equations represent two non-parallel lines in the xy-plane and
the lines intersect at the origin. So the system has only the trivial solution.
(b) No. It is obvious that the linear system has only the trivial sol
Solutions to Tutorial 5
1. There are many different ways to write the answers.
(a) Substituting (x, y) = (1, 2) and (2, 1) into the equation ax + by = c, we has
a system of linear equations
a + 2b c = 0
2a b c = 0
We obtain a = 35 c and b = 15 c. Thus 3x
Solutions to Tutorial 1
1. (There are different ways to write down the general solutions.)
(a) x =
3
4
+ 12 t, y = t where t is an arbitrary parameter.
(b) x1 = 1 + 5s 6t, x2 = r, x3 = s, x4 = t where r, s, t are arbitrary
parameters.
(c) v = 4q r + 21 s