The University of Sydney
School of Mathematics and Statistics
Solutions to Assignment 1
MATH2961: Linear Algebra and Vector Calculus (Advanced)
Semester 1, 2013
Lecturers: Daniel Daners and Zhou Zhang
The assignment is due on Thursday 28th March, at the e
The University of Sydney
School of Mathematics and Statistics
Vector Calculus Assignment
MATH2061/2067: Vector Calculus
Semester 1, 2016
This assignment is due by Friday 20 May 2016 at 4:00pm
and is worth 5% of your assessment for Vector Calculus.
Submit
The University of Sydney
School of Mathematics and Statistics
Vector Calculus Quiz Preparation Sheet
MATH2961: Linear Algebra and Vector Calculus (Advanced)
Semester 1, 2013
Family name:
Other names:
SID:
Signature:
Please note:
(a) The quiz runs for 40 m
The University of Sydney
School of Mathematics and Statistics
Assignment
MATH2962: Real and Complex Analysis (Advanced)
Semester 1, 2013
Web Page: http:/www.maths.usyd.edu.au/u/UG/IM/MATH2962/
Lecturer: Florica C
rstea
Due on Thursday, 9 May by 5pm in Car
The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH2961: Linear Algebra and Vector Calculus (Advanced)
Semester 1, 2013
Lecturers: Daniel Daners and Zhou Zhang
The assignment is due on Thursday 28th March, at the end of the lec
The University of Sydney
School of Mathematics and Statistics
Assignment 2 (Linear Algebra)
MATH2961: Linear Algebra and Vector Calculus (Advanced)
Semester 1, 2013
Web Page: http:/www.maths.usyd.edu.au/u/UG/IM/MATH2961/
Lecturers: Daniel Daners and Zhou
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 1 (Week 2)
MATH2961: Linear Algebra and Vector Calculus (Advanced)
Semester 1, 2013
Lecturers: Daniel Daners and Zhou Zhang
Topics covered and aims
In lectures last week:
WEEK 7
Summary of week 7 (lectures 19, 20 and 21)
Recall that if A is an np matrix over the eld R then the column space of A,
which by denition is the subspace of Rn consisting of all linear combinations of
the columns of A, is given by
CS(A) = cfw_ Au |
WEEK 8
Summary of week 8 (Lectures 22, 23 and 24)
This week we completed our discussion of Chapter 5 of [VST].
Recall that if V and W are inner product spaces then a linear map T : V W
is called an isometry if it preserves lengths; that is, T is an isomet
WEEK 5
Summary of week 5 (lectures 13, 14 and 15)
Lectures 13 and 14 saw the conclusion of Chapter 4 of [VST]. First, Theorem 3.15 was proved, then Theorem 4.18. (The proofs can be found in [VST].) It
follows from Theorem 4.18 that if v1 , v2 , . . . , vn
WEEK 6
Summary of week 6 (lectures 16, 17 and 18)
Every complex number can be uniquely expressed in the form = a + bi,
where a, b are real and i = 1. The complex conjugate of is then dened to
be the complex number a bi. We write for the complex conjugate
WEEK 4
Summary of week 4 (lectures 10, 11 and 12)
We have seen that if F is any eld then F n , the set of all n-tuples of scalars,
is a vector space over F . The aim of this weeks lectures was, roughly speaking,
to show that there are essentially no other
WEEK 9
Summary of week 9 (Lectures 25, 26 and 27)
Lecture 25 and the rst part of Lecture 26 were concerned with permutations.
See Denitions 8.1, 8.2 and 8.3 of [VST]. The notation we use is that introduced
on p. 171 of [VST]. See page 173 for examples of
WEEK 11
Summary of week 11 (Lectures 31, 32 and 33)
The contents of this weeks lectures coincided almost exactly with Chapter 7
This involved repeating some of the material from Week 5.
Recall that if V is an n-dimensional vector space over F and b is a b
WEEK 10
Sylows Theorem
The term modern algebra principally refers to abstract theories in which the objects
of study are assumed to satisfy certain basic rules, or axioms, but are otherwise undened.
Its prominence stems from the discovery of many examples
WEEK 12
Summary of week 12 (Lectures 34, 35 and 36)
We have dened the right null space of A Mat(m n, F ) to be the space
of all v F n such that Av = 0. We call the dimension of the right null space of
A the right nullity of A. Similarly, the left null spa
STAT2011 Statistical Models
sydney.edu.au/science/maths/stat2011
Semester 1, 2015
Lecturer: Michael Stewart
Tutorial Week 2
1. Suppose two 6-sided dice (say a green one and a red one) are rolled in such a way that all
36 possible outcomes are equally like
STAT2011 Statistical Models
sydney.edu.au/science/maths/stat2011
Semester 1, 2015
Lecturer: Michael Stewart
Tutorial Week 3
1. Suppose Graham and Rebecca each ip two coins independently of one another in such
a way that all 4 possible outcomes cfw_TT , HT
STAT2011 Statistical Models
sydney.edu.au/science/maths/stat2011
Semester 1, 2015
Lecturer: Michael Stewart
Tutorial Week 1
Consider an experiment where 3 coins are ipped in sequence. There are 8 = 23 possible
outcomes (assuming the coin only lands heads
The University of Sydney
School of Mathematics and Statistics
Solutions to Practice session 10 (Week 6)
MATH2061: Vector Calculus
Summer School
1. Find F in each of the following:
(a) F = i + j + k
x
y
i+
(b) F =
j
x2 + y 2
x2 + y 2
(c) F = x i + y j + z
WEEK 2
Summary of week 2 (lectures 4, 5 and 6)
Lecture 4 was concerned with the concept of linearity. This material appears
3a of the text Vector Space Theory (referred to as [VST] below).
Denition Let V and W be vector spaces over the same eld F . A func
2
Applying the same proposition with C in place of B gives
The University of Sydney
MATH2902 Vector Spaces
(2)
(http:/www.maths.usyd.edu.au/u/UG/IM/MATH2902/)
Semester1, 2001
Lecturer: R. Howlett
Since CS(B) = CS(C) the right hand side of (1) equals the r
2
of R2 and R.
(ii ) With as in (i) and as in Exercise 1 calculate and its matrix relative
the two given bases. Hence verify that Mbd () = Mbc () Mcd ().
The University of Sydney
MATH2902 Vector Spaces
(http:/www.maths.usyd.edu.au/u/UG/IM/MATH2902/)
Semes
2
We nd that
The University of Sydney
MATH2902 Vector Spaces
(http:/www.maths.usyd.edu.au/u/UG/IM/MATH2902/)
t
AA =
Semester1, 2001
Lecturer: R. Howlett
(1, 2),
(2, 2),
1
1
1
2
1
2
t
A2 =
5
5
and so it follows that
Let y = a + bx be the equation of t
The University of Sydney
School of Mathematics and Statistics
Practice session 10 (Week 6)
MATH2061: Vector Calculus
Summer School
1. Find F in each of the following:
(a) F = i + j + k
y
i+
(b) F =
x2 + y 2
(c) F = x i + y j + z k
x
x2 + y 2
j
2. Find the
The University of Sydney
School of Mathematics and Statistics
Solutions to Practice session 7 (Week 5)
MATH2061: Vector Calculus
1. Evaluate
(2, 2, 5).
Solution:
t : 0 1.
Summer School
ds if = xy + z and C is the straight line segment from (1, 1, 1) to
C
The University of Sydney
School of Mathematics and Statistics
Solutions to Practice session 9 (Week 6)
MATH2061: Vector Calculus
Summer School
1. Sketch the following regions:
(a) cfw_(x, y) R2 | 1 x 2, x y x
(b) cfw_(x, y) R2 | x = r cos , y = r sin , 0
The University of Sydney
School of Mathematics and Statistics
Practice session 7 (Week 5)
MATH2061: Vector Calculus
1. Evaluate
Summer School
ds if = xy + z and C is the straight line segment from (1, 1, 1) to
C
(2, 2, 5).
2. Describe geometrically the v