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 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 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 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 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
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 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 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
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 2
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 4 (31/8
- 4/9).
You are advised to revise Sections 1.4 to
Mid Term Test
1) Date: 2nd March (Monday of Week 7)
2) Time: 4.15pm - 5.15pm
3) Venue: MPSH1 Section B.
4) Seating plan (according to tutorial groups) will
be uploaded later this week.
5) Scope: Everything up to and including Section 3.2.
6) Helpsheet: (n
MA1101R
Linear Algebra I
Introductory Lecture (Lecture 00)
S
Welcome
S All NUS students
S All H3 students
S All non-NUS (exchange, NUS High) students
Outline of todays lecture
S Part I: Module information
S Part II: About Linear Algebra
S Part III: Study
Definition of matrices, entries, size of a matrix.
Special types of matrices.
Matrix operations and some laws.
Matrix multiplication and differences with real
number multiplication.
Different ways to represent matrix multiplicatio
What can row-echelon forms tell us about the
solution set of linear systems.
Standard notations to use when performing
elementary row operations.
Three examples on the 'same' kind of problem.
Homogeneous linear systems and
Matrix'transpose'and'some'properties.
Definition of invertible matrices.
Uniqueness of inverse.
Some laws involving inverses.
Representing an elementary row operation by a matrix.
Definition of an elementary matrix.
Elementary matrices
Homework 2 will be uploaded before the weekend
Homework 2 will be due on 16th February
Homework 1 will be returned during tutorial next week
Equivalent statements to 'A is invertible'.
One way to determine if A1 exists
Chapter 1
Linear Systems and Gaussian
Elimination
Chapter 1 Linear Systems and Gaussian Elimination
Section 1.1
Linear Systems and Their
Solutions
Lines in xy-plane
(Discussion 1.1.1)
A line in the xy-plane can be
represented algebraically by
an equation
National University of Singapore
Department of Mathematics
Semester 1, 2015/16
MA1101R Linear Algebra I
Tutorial 1
Questions of this tutorial sheet will be discussed in the tutorial classes in Week 3 (24/8
- 28/8).
You are advised to revise Sections 1.1 t
Solutions to Tutorial 1
2
1. (a) x = t, y = 5 t where t is an arbitrary parameter.
(b) x = r, y = s, z = t, w = 1 (1.5 + 2r + 5s 6t) where r, s, t are arbitrary
8
parameters.
(c) x2 = a, x3 = b, x4 = c, x5 = d, x1 = 1 (1 + 8a 2b c + 4d) where a, b, c, d
3
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.
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
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
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
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
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 2, 2016/2017
MA1101R Linear Algebra
Homework Assignment 2
Please write your name, student card number and tutorial group number on the
answer script, and submit during the lecture on 16th February 2017 (Thursday).
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 2, 2016/2017
MA1101R Linear Algebra
Homework Assignment 1
Please write your name, student card number and tutorial group number on the
answer script, and submit during the lecture on 2nd February 2017 (Thursday).
Solutions to Homework 4
1. (a) (i) Q has no zero row if and only if spancfw_u1 , u2 . . . , uk = Rn .
Reason: See Discussion 3.2.5.
(ii) Q has no non-pivot column if and only if u1 , u2 , . . . , uk are linearly
independent.
Reason: Consider the vector e
Solutions to Tutorial 8
1. (a) Since
(x1 , x2 , x3 , x4 )T = (t 2s, s + t, s, t)T = s(2, 1, 1, 0)T + t(1, 1, 0, 1)T ,
cfw_(2, 1, 1, 0)T , (1, 1, 0, 1)T is a basis for the nullspace of A.
The nullity of A is 2.
(b) A general solution of Ax = b is
x1 = t 2
Solutions to Homework 2
1. (a) Method 1: (Use Theorem 2.5.15.)
The determinant of the matrix
a
0
0
0
is
1
b
0
0
0
1
c
0
= abcd.
0
0
1
d
The matrix is invertible if and only if its determinant is nonzero, i.e. a 6= 0,
b 6= 0, c 6= 0 and d 6= 0.
Method