The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
5.
By the formula for 2 2 matrices, the inverse of
x y
z w
6.
2014
Longer Solutions to Selected Exercises for Week 10
=
11 4
15 5
4 3
7 5
is
4 3
7 5
1 1
2 3
.
5 3
7 4
=
, so that
The m
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
2014
Exercises for Week 10
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a credit or
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
9.
Longer Solutions to Selected Exercises for Week 3
2014
(ii) Observe that P Q = SR = 2i 2j k so P QRS is a parallelogram. But
|P Q| = |P S| = 3 ,
so P QRS is a rhombus. This rhombu
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
11.
Longer Solutions to Selected Exercises for Week 9
2014
The matrix equation A0 = 0A = 0 makes sense by interpreting the symbol 0 as an
abbreviation for zero matrices of compatible d
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
7.
Longer Solutions to Selected Exercises for Week 4
2014
(i) First observe that QP = 2i + 2j + k and QR = i + 2j + 2k , yielding
QP QR = 2 + 4 + 2 = 8 > 0 ,
so that P QR is acute
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
2014
Exercises for Week 9
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a credit or
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
Exercises for Week 4
2014
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a credit or
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
1.
(i)
2014
Longer Solutions to Selected Exercises for Week 8
1
1
2 3
6
2
1
1
0 5
6
10
1 1
0 1
6
2
1 0
0 1
4
2
,
so that x = 4 and y = 2 .
2.
3.
(ii) By back substitution, z = 2 , y =
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
2014
Exercises for Week 8
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a credit or
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
Longer Solutions to Selected Exercises for Week 7
2014
1.
Making each expression equal to 0 produces the point P (2, 1, 4), and equal to 1
produces the point Q(5, 1, 5). A vector in th
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
Exercises for Week 7
2014
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a credit or
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
5.
Longer Solutions to Selected Exercises for Week 5
We have
QP QR =
2i + j + 3k i + 2j k
Hence the area of the triangle P QR is
7i 5j 3k
=
2
6.
2014
= 7i + 5j + 3k .
83
.
2
(i) Obse
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
2014
Exercises for Week 5
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a credit or
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
Exercises for Week 3
2014
Preparatory exercises ideally should be attempted before coming to the tutorial. A suggestion
is given for exercises to be completed during and after the tuto
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
Exercises for Week 2
2014
There are thirteen teaching weeks. There will be one set of exercises each week from Week
2, except for Weeks 6 and 11, when quizzes will be held during tutor
The University of Sydney
MATH1902 Linear Algebra (Advanced)
Semester 1
Longer Solutions to Selected Exercises for Week 2
11.
2014
6
8
d
By Pythagoras d = 82 + 62 = 10 . If is the angle to the horizontal then cos =
6/10 , yielding an angle 53 . Thus the re