NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2011-2012 (Semester 2)
MA1101R
Tutorial Set 1 Solutions
1. (a) True. Since there are three rows in the augmented matrix, the system has 3
equations. Since there are four columns in the augmented m
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
2011-2012 (Semester 2)
MA1101R
Tutorial Set 2 Solutions
1. (a) True. Let A = (aij )nn and B = (bij )nn . Since aij = bij = 0 for i = j, the
(i, j)-entry of AB is equal to
aii bii if i = j
0
if i =
MA1101R Tutorial Note 1, AY 2010/2011
1
Welcome Message
Thank you for choosing to study in this tutorial group! I hope your time here is an enjoyable and rewarding
experience. This note has been compiled to help you understand the textbook and tutorial qu
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1101R Laboratory 2
Semester I 2013/14
Solving systems of linear equations and Elementary Matrices
In this laboratory session, we will investigate systems of linear equations and their
solutions.
Answers/Solutions of Exercise 7
(Version: May 25, 2012)
1. (a) T1 is a linear transformation with standard matrix
1 1
.
1 1
(b) T2 is not a linear transformation.
1 1
(c) T3 is a linear transformation with standard matrix 0 0.
0 0
(d) T4 is not a linear t
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
Answers/Solutions of Exercise 4
(Version: May 25, 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), we
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1101R Laboratory 5
Semester I 2013/14
More Eigenvalues, Diagonalization
and Orthogonal Projection
In this laboratory session, we will continue to use matlab to solve problems involving eigenvalu
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1101R Laboratory 3
Semester I 2013/14
Linaer Combinations, Linear Span, Linear Independence,
Transformation Matrices
In this laboratory session, we will consolidate what we have learnt in lectur
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1101R Laboratory 3
Semester I 2013/14
Linaer Combinations, Linear Span, Linear Independence,
Transformation Matrices
In this laboratory session, we will consolidate what we have learnt in lectur
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1101R Laboratory 4
Semester I 2013/14
Transition matrices, Row space, Nullspace
Eigenvalues and Diagonalization
In this laboratory session, we rst examine an example on transition matrices. This
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1101R Laboratory 5
Semester I 2013/14
More Eigenvalues, Diagonalization
and Orthogonal Projection
In this laboratory session, we will continue to use matlab to solve problems involving eigenvalu
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA1101R Laboratory 4
Semester I 2013/14
Transition matrices, Row space, Nullspace
Eigenvalues and Diagonalization
In this laboratory session, we rst examine an example on transition matrices. This