T HE U NIVERSITY OF S YDNEY
P URE M ATHEMATICS
Linear Mathematics
2016
Tutorial 4 (Preliminary and extra exercises) Solutions
1. In each of the following find a basis for Span(X), and say whether or not Span(X) is equal to
the vector space
Rn of which X o

T HE U NIVERSITY OF S YDNEY
P URE M ATHEMATICS
Linear Mathematics
2016
Tutorial 2 (Preliminary and extra exercises) Solutions
1.
n
o
a) Explain why the set A =
| x R is not a subspace of R2 .
o
n
1
1
0 +t 1
| t R is not a subspace of R3 .
b) Explain wh

T HE U NIVERSITY OF S YDNEY
P URE M ATHEMATICS
Linear Mathematics
2016
Tutorial 3 (Preliminary and extra exercises) Solutions
1 2 3
2 5 4
1. Let A =
.
a) Determine Null(A) geometrically.
b) Show that Col(A) = R2 .
c) Write each of the vectors 10 and 01

The University of Sydney
School of Mathematics and Statistics
Solutions to Exercises 7
MATH2061: Vector Calculus
Summer School
1. Let u = 3i j k, v = i + 3j + k and w = 2i 3k. Find
(u v)w, u(v w), (u v) w, u (v w), (u v) w, u (v w).
Solution:
uv = 3(1)+(1

The University of Sydney
School of Mathematics and Statistics
Solutions to Exercises 9
MATH2061: Vector Calculus
1. Let I =
ZZ
Summer School
xy dxdy where R is the region in the xy plane bounded by y = x2 ,
R
x = 0, y = 4 and where x 0.
(a) Sketch the reg

T HE U NIVERSITY OF S YDNEY
P URE M ATHEMATICS
Linear Mathematics
2016
Tutorial 1 (Preliminary and extra exercises) Solutions
1.
5
3 0 4
2.
1 1 2
a) Calculate the matrix product
4
1 3 5
b) Hence find all solutions to the system of linear equations:
3x1

The University of Sydney
School of Mathematics and Statistics
Solutions to Exercises 8
MATH2061: Vector Calculus
Summer School
1. Find grad f if f (x, y) = x2 cos xy.
Solution:
grad f = f =
f
f
i+
j = (2x cos xy x2 y sin xy) i x3 sin xy j.
x
y
2. Find gra

T HE U NIVERSITY OF S YDNEY
P URE M ATHEMATICS
Linear Mathematics
2016
Tutorial 5 (Preliminary and extra exercises) Solutions
0 1
is not diagonalisable.
1. Show that the matrix
0 0
1
= 2 , so the only eigenvalue is zero.
Solution
0
n
o
t |t R , whic

The University of Sydney
School of Mathematics and Statistics
Vector Calculus Assignment
MATH2061: Vector Calculus
Summer School, 2016
This assignment counts for 5% of your overall assessment for Vector Calculus.
You should submit this assignment by givin

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