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
Tutorials and WebAssign Workbooks
Tutorials
Tutorials will be held next week in Week 7.
There will be no tutorials in Week 8, after the lecture break.
WebAssign Workbooks
Your tutor will check your WebAssign workbooks during the tutorial in Week 7.
Gri Wa
Example From Previous Lecture
An example of a parametric equation for a line is
x
1
2
x = p + tv
where x = 1 , p =
and v =
.
x2
0
1
Every point on the line corresponds to a value of t, and vice-versa.
x2
rr
2 T
r
rr
s
r t = 1.5
HH
rr
x = p + tv
HH x r
r
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
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
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
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): grith.ware@anu.edu.au
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
(component-wise addition)
v
u+
u
v
u+
u
(parallelogram)
or:
v
a1
a2
Graphical Interpretation
v
Algebraic De
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 ).
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 mid-semester exams.
WebAssign Quizzes have begun again - the remaining Quizzes 6 to 10 are
Important points from yesterday
Matrices of the same size can be added together and multiplied by scalars
component-wise.
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
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 in-tutorial 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
Paper-based 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 non-zero 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
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
Limit of a function at a point
Limits at infinity
Week 2: Limit of a Function
July 23, 2015
Week 2: Limit of a Function
Limit of a function at a point
Limits at infinity
1
Limit of a function at a point
2
Limits at infinity
Lecture 4
Week 2: Limit of a Fu
Linear Approximation
Maxima and Minima
Week 5: Maxima and Minima, Mean Value
Theorem
August 12, 2015
Week 5: Maxima and Minima, Mean Value Theorem
Linear Approximation
Maxima and Minima
1
Linear Approximation
2
Maxima and Minima
The Mean Value Theorem
Der
MATH1013
Mathematics and Applications I
ALGEBRA section
(Linear Algebra and Complex Numbers)
Stephen Roberts
PAP Moran Building, Room 2005
Ph: 6125 4445
eml: stephen.robertsy@anu.edu.au
Semester 2, 2016
Stephen Roberts (ANU)
ANU MATH1013 Algebra
Semester
MATH1013
Mathematics and Applications I
ALGEBRA section
(Linear Algebra and Complex Numbers)
Stephen Roberts
PAP Moran Building, Room 2005
Ph: 6125 4445
eml: stephen.robertsy@anu.edu.au
Semester 2, 2016
Stephen Roberts (ANU)
ANU MATH1013 Algebra
Semester
MATH1013
Mathematics and Applications I
ALGEBRA section
(Linear Algebra and Complex Numbers)
Stephen Roberts
PAP Moran Building, Room 2005
Ph: 6125 4445
eml: stephen.robertsy@anu.edu.au
Semester 2, 2016
Stephen Roberts (ANU)
ANU MATH1013 Algebra
Semester
BUSN3001 WEEK 9 TUTORIAL QUESTIONS
1. Textbook Q10.9 (p532): If individuals have access to insider information and are
able to make large gains on a securities market as a result of using information that
is not widely known, then is this an indication th
Brett Parker / Stephen Roberts
Department of Mathematics
Australian National University
Semester 2, 2016
MATH 1013 - Mathematics and Its Applications
Tutorial Worksheet 2
1
Algebra
a). Lay, 3rd Ed., Page 48, Q 22.
Let
0
v1 = 0 ,
2
0
v2 = 3 ,
8
4
v3 = 1 .
Continuity
Derivatives
Week 3: Limits, continuity, derivatives of functions
July 30, 2015
Week 3: Limits, continuity, derivatives of functions
Continuity
Derivatives
1
Continuity
Continuous functions
The intermediate value theorem
2
Derivatives
Meaning an