Department of Mathematics
MATH1003
TUTORIAL 5 : Week 8
This weeks assignment is questions 3 and 6.
1. The logistic model below represents the population (in grams) of a bacterium after t
hours.
1800
.
P (t) =
1 + e(t10)
(a) What is the carrying capacity C
Tutorial 4 (Week 6)
General review: Determinant
1. What is a determinant?
Def. 2.5.2, Not. 2.5.3 (Original definition)
Thm 2.5.6 (Cofactor expansion, equivalent definition)
2. Properties of the determinant
a) For matrices with special structures
Thm 2.
Linear Algebra beyond Mathematics
August 25, 2016
Linear algebra applies to various fields in natural and social sciences. This article picks up
three of these applications in the field of economics (input-output model), statistics (time series
analysis)
Department of Mathematics
MATH1003
TUTORIAL 3 : Week 5
1. Find all polynomials in t of degree 2 or less whose graphs pass through the following
points: (1, 1), (2, 3).
2. Adam, Bob and Cathy purchased biscuits of different brands P, Q and R. Adam purchase
assignment 2: 3(b) 3(c) Q4
Department of Mathematics
MATH1003
TUTORIAL 2 : Week 4
1. A Furniture Factory has 1950 machine hours available each week in the cutting department, 1490 hours in the assembly department, and 2160 hours in the finishing
departmen
next week's tute assignment: Q7 & Q11
Department of Mathematics
MATH1003
TUTORIAL 1 : Questions
1. Derive the distance formula between two points in R3 :
P = (x1 , y1 , z1 ),
and
Q = (x2 , y2 , z2 )
2. Find the slope of each line.
slope is undefined, it i
Q1
/18
Q2
Q3
/10
Q4
/8
/15
Q5
/10
Q6
/10
Q7
/9
Q8
Q9
/10
Total
/10
The Australian National University
Final Exam - June 2015
MATH1003 - Algebra and Calculus Methods.
Student No.:
u
Important notes:
You must justify your answers. Do not expect credit for
Q1
/15
Q2
/10
Q3
/10
Q4
/10
Q5
Q6
/10
/10
Total
/65
The Australian National University
March 2014
MATH1003 - Algebra and Calculus Methods
MID-SEMESTER EXAMINATION
Student No.:
u
Important notes:
One A4 sheet handwritten both sides permitted.
No calculat
The Australian National University
March 2014
MATH1003 — Algebra and Calculus Methods
MID—SEMESTER EXAMINATION
Student No.2
Important not es:
0 One A4 sheet handwritten both sides permitted.
0 No calculators or books permitted.
0 You have 15 minutes r
Australian
9%9 National
Unnversnty
The Australian National University
Final Exam — June 2015
MATH1003 — Algebra and Calculus Methods.
Student N0.:
Important notes:
0 You must justify your answers. Do not. expect credit for a correct answer with no
justi
Answers/Solutions of Exercise 1
(Version: October 2, 2016)
1. (a), (c), (f), (i), (l) are linear equations and (b), (d), (e), (h), (j) are not.
For (g), although the equation does not look like a linear equation, if we apply
the logarithmic function to bo
Answer to exercise problems in the review
September 15, 2016
Dear friends,
Below please kindly find my answer to some exercise problems covered in the review list for
Tutorial 4, in case you are curious about them.
Question 1. Factorize the following four
Tutorial 3 (Week 5)
General review (Matrix inverse: The 8th operation of matrices)
1. What is matrix inverse: Def. 2.3.2, Not. 2.3.6, Def. 2.3.11
2. How to depict matrix inverse?
a) Thm 2.3.5 (Uniqueness)
b) Thm 2.3.9 (Arithmetics)
Attention:
(Generally:
Advice about writing a concise proof
September 8, 2016
Dear friends,
As mid-term is drawing near, I would love to share with you how to make your answer
accessible to your grader.
Take writing a concise proof for instance. As far as I see, the following p
A rigorous answer to Question 6(b), Tutorial 1
August 24, 2016
Dear friends,
Below please kindly find a rigorous proof of Question 6(b) with referring to Theorem 1.2.7
from your textbook. Typical details worth paying attention to have been highlighted wit
Answers/Solutions of Questions 1-14 of Exercise 3
(Version: September 10, 2016)
1. u = (1, 3), v = ( 3, 1), u + v = (1 3, 1 + 3),
3u 2v = (3 + 2 3, 2 + 3 3).
2. See Question 1 of Tutorial 5.
3. A = B = C = F and A, D, E are all different.
4. (a) U and V c
Tutorial 5 (Week 7)
General review: Preliminaries to vector space (Secs. 3.1-3.4)
A. Recap: Why to focus on the vector space?
a) Rmk 1.1.10 (three possibilities of the solution set of a linear system)
b) Thm 2.4.7-2 (sufficient and necessary condition for
Clarification for the exercise problem in class
September 1, 2016
Dear friends,
I mentioned in class an exercise problem attached to Question 2(b), which goes
Question 1. Provided that [A, B] = AB BA for any square matrices A and B of the same
size, show
A rigorous answer to Question 5, Tutorial 2
September 1, 2016
Dear friends,
Below please kindly find the rigorous proof of Question 5 (using two different methods).
Proof (Method 1). Only need to show that u and v may generate infinitely many solutions to
Differences between T5-Q6 and HW2-Q6
October 1, 2016
Dear friends,
Question 6 in Tutorial 5 and in Homework 2 resemble each other. They have similar known
conditions:
(1) Column vectors u1 , u2 , ., uk Rn and v1 , v2 , ., vk Rn ; and that
(2) Column vecto
Marking Scheme of the Mid-Term Test
1. Use the Gaussian Elimination or Gauss-Jordan Elimination to solve the following linear
system:
+ 2z = 0
w x
w x + 4y 2z = 4a
2w 2x + 4y + az = b
where a and b are constants. Indicate the elementary row operation use
Tutorial 1 (Week 3)
A. Introduction
Linear algebra Methods for solving linear systems
Typical linear systems
System of algebraic linear equations (Def. 4.3.1 + Thm 4.3.6)
Linear recurrence equation (Thm 6.2.3 + Alg. 6.2.4 + Dsn 6.2.10)
System of line