Check of WebAssign Workbooks in Week 13 Tutorials
Reminder:
Tidy up your WebAssign Workbook and bring it to
your tutorial next week (26th to 30th May).
You should include your rough working for each of the WebAssign
Quizzes 5 through 10, legibly presented
MATH1013
Mathematics and Applications I
Algebra Section
(Linear Algebra and Complex Numbers)
Gri Ware
PAP Moran Building, Room 1004
Ph: 6125 2431
email (Mon Thu): [email protected]
Any application of the ANU logo on
a coloured background is subject
to
Addition and Scalar Multiplication of Vectors
The following is for R2 , with the obvious extension to R3 and beyond.
Sum:
=
+
b1
b2
a1 + b1
a2 + b2
(componentwise addition)
v
u+
u
v
u+
u
(parallelogram)
or:
v
a1
a2
Graphical Interpretation
v
Algebraic De
Linear equations
Denition (Linear equation)
A linear equation in variables x1 , . . . , xn is an equation of the form
a1 x1 + a2 x2 + + an xn = b,
where ai and b are scalars (real numbers or complex numbers).
Gri Ware (ANU)
ANU MATH1013 Algebra
Semester 1
Techniques of integration
Week 13: Techniques of integration
May 27, 2014
Week 13: Techniques of integration
Techniques of integration
1
Techniques of integration
Trigonometric integrals
The method of partial fractions
Lecture 23
Week 13: Techniques of in
Inverse functions
Indeterminate forms
Integration techniques
Week 11: The Inverse Trigonometric Functions,
Hyperbolic Functions, Indeterminate Forms
May 13, 2014
Week 11: The Inverse Trigonometric Functions, Hyperbolic Fun
Inverse functions
Indeterminate
General exponential and logarithmic functions
Week 10: Exponential functions, Exponential
growth
May 2, 2014
Week 10: Exponential functions, Exponential growth
General exponential and logarithmic functions
The derivative of e x .
What is the derivative of
The natural logarithm
The natural logarithm & exponential
General exponential and logarithmic functions
Week 9: The natural logarithm and exponential
functions
April 24, 2014
Week 9: The natural logarithm and exponential functions
The natural logarithm
Th
Inverse functions
Week 8: Inverse functions
April 23, 2014
Week 8: Inverse functions
Inverse functions
1
Inverse functions
Inverse functions and calculus
Week 8: Inverse functions
Inverse functions
Recall from last class
The substitution rule
If u = g (x)
Fundamental Theorem of Calculus
Week 7: Fundamental Theorem of Calculus
March 31, 2014
Week 7: Fundamental Theorem of Calculus
Fundamental Theorem of Calculus
1
Fundamental Theorem of Calculus
FTC and applications
Evaluating integrals
Lecture 14
Week 7: F
Limits of functions at a point
Limits at innity
Week 2: Limit of Functions
February 26, 2014
Week 2: Limit of Functions
Limits of functions at a point
Limits at innity
1
Limits of functions at a point
2
Limits at innity
Lecture 4
Week 2: Limit of Function
Advertisement:
Morning Tea for Women Interested in Mathematics
The morning tea is for women interested in mathematics, and is in the common
room on the 2nd oor of the John Dedman Building (just next to Manning Clark
lecture theatres: go through the doors
Two questions
1. Can a linear system with a 2 3 coecient matrix have exactly one solution?
2. Can a linear system with a 3 2 coecient matrix have innitely many
solutions?
If you have an internet connection here in the lecture, please answer these
question
8. Linear Transformations (Parts of Lay 1.8 and 1.9)
Case Study: http:/media.pearsoncmg.com/aw/aw_lay_linearalg_3/cs_
apps/lay03_02_cs.pdf
Lays Summary: http:/media.pearsoncmg.com/aw/aw_lay_linearalg_3/
parachute/tm/c01/pdf_files/sec1_8ov.pdf and
http:/me
Check of WebAssign Workbooks in Week 13 Tutorials
Reminder #1:
Tidy up your WebAssign Workbook and bring it to
your tutorial this week.
You should include your rough working for each of the WebAssign
Quizzes 5 through 10, legibly presented and organised b
Remaining Assessment Items
Week 12 and Week 13 intutorial Quizzes and Worksheets.
WebAssign Quiz 10 (due this coming Sunday 25th May).
Check of WebAssign Workbooks in Week 13 Tutorial.
Dont forget to tidy up your WebAssign Workbooks and
bring them to you
Factorising A as LU
We saw in Section 11 that applying row operations to A is equivalent to
multiplying A by elementary matrices.In particular, if U is an echelon form for A
then there are elementary matrices E1 , E2 , . . . , Ep such that
Ep E1 A = U
the
Final Exam
The Final Exam will be held:
on Friday, 6 June 2014,
from 14:15 to 17:30
(15 min reading time + 3 hr exam).
The venue is the Sports Hall (Bldg 19).
Permitted materials:
A4 page (1 sheet) with handwritten notes on both sides
Paperbased English
16. Properties of Determinants (Lay 3.2)
Cramers Rule, Volume and Linear Transformations (Lay 3.3)
Case Study: http:
/media.pearsoncmg.com/aw/aw_lay_linearalg_3/cs_apps/jacobian.pdf
This will be the last lecture based on content in Lays Linear Algebra tex
From Previous Lecture
Denition
The column space of a matrix A is the subspace Col A = Span cfw_a1 , . . . , an of
all linear combinations of the columns a1 , . . . , an of A.
Denition
The null space of a matrix A is the subspace Nul (A) of all solutions
From Previous Lecture
Theorem (B: Invertibility Characterisations 3 and 4)
For an n n matrix A the following are equivalent:
0
A is invertible.
3
There is an n n matrix C with CA = In . [Statement (j) in Lay Thm. 8]
4
There is an n n matrix D with AD = In
Elementary Matrices
Recall the denition of elementary row operations:
Add a multiple of one row to another row.
Interchange two rows.
Multiply a row by a nonzero constant.
Denition
A matrix obtained from an identity matrix by a single elementary row oper
Previous Lecture  Standard Matrix
Theorem (THEOREM 10 in Lay)
Let T : Rn Rm be a linear transformation. Then there exists a unique m by n
matrix A such that
T (x) = Ax for all x Rn .
In fact A is the matrix whose jth column is the vector T (ej ), where e
Important points from yesterday
Matrices of the same size can be added together and multiplied by scalars
componentwise.
An m k matrix A and an l n matrix B can be multiplied together to give
the m n matrix product AB only when k = l. Otherwise the produ
Admin
No tutorials are running this week. Tutorials resume next week (Week 9).
In your Week 9 tutorials, you will be given the opportunity to check over your
marked midsemester exams.
WebAssign Quizzes have begun again  the remaining Quizzes 6 to 10 are
Linear Transformations
All matrix transformations have an important property which comes from similar
properties of matrices noted previously they are linear.
Denition
A transformation T : Rn Rm is linear if
1
T (u + v) = T (u) + T (v), (for all u,v Rn ).
Power series  introduction
Two of the many uses of power series are to represent alreadyknown
functions and to define new ones. Since the power series may not converge
everywhere the function is defined, we must be aware of the fact that the
series may
Q5 oml
The Australian National University
Final Exam  November 2010
MATH1013  Mathematics and its Applications 1
Section B Algebra
Student No.: SOLO TIC) MS
Important notes:
You must justify your answers. Be neat.
One A4 sheet hand written both si
Australian
a National
MW . f"University
Final Exam  November 2014
MATHlOlB Mathematics and its Applications 1.
Book 13 Algebra
Questions 1 8
Student No.1
Imp ortant notes:
0 You must justify your answers. Do not expect credit for a correct answer
with
.
Australian
a 9 National
Umversnty
Final Exam  June 2014
MATHlOlB Mathematics and its Applications 1.
Book A Calculus
Questions 1 7
Student No:
Import ant notes:
a You must justify your answers. Do not expect credit for a correct answer
with n
Q1
Q2
Q3
Q4
Total
The Australian National lJniversity
Final Exam  November 2010
MATH1013  Mathematics and its Applications I
Part A
Calculus

Student
No.:
5"\^tr,\
t
Important notes:
o You rnust justify
o
Aour" ansuers. Be neat.
One .A4 sheet ha,nd wr