COURSE OUTLINE
MATH1031
MATHEMATICS FOR LIFE SCIENCES
Semester 1, 2016
Cricos Provider Code: 00098G Copyright 2016 -School of Mathematics and Statistics, UNSW Australia
Welcome to MATH1031
REFERENCES FOR MATH1031 (LEC 1 WED/THU)
WEEK
MATERIAL TO BE COVERE
Applications of Systems with Unique Solutions
Quick Review
In a row echelon form for a system of linear equations,
if the right-hand column is leading, there is no solution.
if all columns of the LHS are leading, but the right-hand column is not, the
sys
Matrix Transformations
Here we will introduce the idea of a matrix transformation. We will discuss some
uses of matrix algebra and matrix transformations.
Constructing and Transforming Images
Suppose we have a triangle in 2.
The vertices as column
vectors
Systems of Linear Equations with Non-Unique Solutions
In a row echelon form of the augmented matrix of a system of linear equations,
if the right-hand column is leading, there is no solution.
if all columns of the LHS are leading, but the right-hand colu
Answers (Applications of Systems with Non-Unique Solutions)
Answers to Questions in
Applications of Systems with Non-Unique Solutions
p.1
Suppose the couple invests $t in Treasury Bills, $c in Corporate Bonds, and $j in Junk
Bonds.
t + c + j = 25000
0.07
Applications of Systems with Non-Unique Solutions
Problem 1.
A retired couple has $25000 available to invest. They require a return on their
investment of $2000 in the first year. As their financial consultant, you recommend
they invest some money in Trea
MATH1031 Mathematics for Life Sciences
Do as much mathematics and statistics as you can in your
degrees - these skills will empower your professional lives
Sir Gustav Nossal, Immunologist
Life sciences are:
quantitative
complex
Mathematics is:
the scie
Matrix Algebra
A matrix is a useful shorthand representation of inter-related quantities.
We can use matrices to simplify and solve problems.
Example:
Suppose we have a delivery service which carries four products: milk, bread butter
and eggs. They record
Markov Processes
Motivation Example.
A psychologist conducts an experiment in which a mouse is placed in a T-maze.
The mouse has a choice at the T junction of turning left and receiving a reward or
turning right and receiving an electric shock.
Reward
Sho
Systems of Linear Equations with Unique Solutions
Solving linear equations by row reduction
We have seen how to represent a system of linear equations by an augmented
matrix and also we learnt how to reduce a matrix into row echelon form. Through
the foll
Answers (Systems of Linear Equations with Unique Solutions)
Answers to Questions in
Systems of Linear Equations with Unique Solutions
p.1
Step 1: Create the augmented matrix.
2 3 1 22
3 2 2 14
1 1
1 0
Step 2: Reduce the augmented matrix to row echel
Answers (Semi-log Plots)
p.2
Answers to Questions in Semi-log Plots
1
b3
3
1. e 3 ln b = eln b = b 3 =
2. Solve for x
ln 3 x 2 = 0
ln 3 x = 2
e ln 3 x = e 2
3x = e 2
1
x = e2
3
3. Apply the exponential function to both sides
ln y = 2.3 x + 7
e ln y = e 2.
Answers (Log-log Plots)
Answers to Questions in Log-log Plots
p.1
1) Y = ln y and x = ln x
a)
y = 2x 5
ln y = ln 2 + 5 ln x
Y = 5 X + ln 2
b)
y=
7
x
= 7x
1
2
1
ln y = ln 7 ln x
2
1
Y = X + ln 7
2
2)
In terms of x, and y:
a)
Y = 6X 2
ln y = 6 ln x 2
e ln y
Log-log Plots
We have seen how to fit an exponential relation to data points in semi-log plot. We
are going to fit a power relation = to the data points.
Revision
1.
For each of the following power relations between y and x find a linear
relation between
Semi-log Plots
Recollection of the log laws
We have seen the following rules.
ln 1 = 0
ln e = 1
ln( xy ) = ln x + ln y
x
ln = ln x ln y
y
ln x n = n ln x
ln(e x ) = x
e ln x = x
MATH1031
Semi-log Plots p.1
Examples.
1. Simplify e 3 ln b .
2. Solve for
Answers (Echelon Form and Row Operations)
Answers to Questions in
Echelon Form and Row Operations
p.1
x1 + 2 x1 x2 + x3 = 2
No, it is not linear, because of the term x1 x2 .
x1 +
1
3x3 = 5 and x1 3 x2 + 2 x3 = 4 are not linear, but
x2
x1 = 2 x3 x2 + 7 ,
Points, Lines and Planes in Space
When we are modelling a system with a dependent variable y in terms of
only one independent variable x , the graph of y against x is an important tool.
What if the dependent variable changes with more than one independent
Answers (Applications of Systems with Unique Solutions)
Answers to Questions in
Applications of Systems with Unique Solutions
p.2
2
1 1
, the first and second columns are leading, but the right-hand column is non
0 1 2
For
leading. Hence, the system h
The number of common points of a set of lines in 2 or a set of planes in 3
can only be (a) none, (b) one, or (c) infinitely many. Algebraically, common points
of a set of lines in 2 are the solutions to the system of linear equations in 2
variables repres
Answers (Systems of Linear Equations with Non-Unique Solutions)
Answers to Questions in
Systems of Linear Equations with Non-Unique Solutions
p.2
Reduce the augmented matrix,
1
2 4 10
3 3 15 15
2 1 1 5
1 2 4 10
R 2 = R 2 + 3R
1
0 9 27 45
R 3 =
Answers (Matrix Algebra)
Answers to Questions in Matrix Algebra
p.3
The entry in the 1st row and 3rd column is A13 = 2 .
The entry in the 2nd row and 1st column is A21 = .
p. 4
The matrices we have looked at in the delivery service example are 3 x 4.
p.7
Answers (Matrix Applications)
Answers to Questions in Matrix Applications
p.3 Problem 1
Oranges Apples
Armidale 20
30
;
A=
Guyra
30
16
8
Uralla
20
Cost
B = Oranges 10.4
Apples 8.8
The column labels of A are the same as the row labels of B.
Quantity
Special Functions
Logarithms and Exponentials
Revision
Laws for Indices.
Assume a, b > 0 , m, n are real numbers.
a m a n = a m+ n
am
= a mn
n
a
(a m ) n = a mn
a mb m = (ab) m
am a
=
bm b
m
MATH1031
Special Functions p.1
Furthermore, we define
a0 = 1
UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1031 Mathematics for Life Sciences 2013
Test 2
Version 2
This sheet must be lled in and stapled to the front of your answers
Students Surname
Given Name or Initials
Tutorial Code
Stude
UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1031 Mathematics for Life Sciences 2013
Test 1
Version 1
This sheet must be lled in and stapled to the front of your answers
Students Surname
Given Name or Initials
Tutorial Code
Stude
UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1031 Mathematics for Life Sciences 2012
Test 1
Version 2
This sheet must be lled in and stapled to the front of your answers
Students Surname
Tutorial Code
Given Name or Initials
Stude
UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1031 Mathematics for Life Sciences 2013
Test 2
Version 7
This sheet must be lled in and stapled to the front of your answers
Students Surname
Given Name or Initials
Tutorial Code
Stude
C E TS E T
HA HE
MATH1051
Polar Form
Angle Between Vectors
Calculus & Linear Algebra
z = rcis = r(cos + i sin )
University of Queensland
Where r is the modulus |z| and the argument arg z .
1
Polar Form Operations and Properties
ab
a b
cos =
Complex Number