University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Example:
Objectives
01
1 0
A=
(1)
Diagonalisation.
What can go wrong?
1
A I =
n
(2)
Calculate
(3)
Solve systems of linear dierential equations
A
( n N)
y = Ay
1
= 2 + 1 = 0
A has eigenvalues which are not real.
3
1
A=
Lecture 17 Applications of Diagon
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
Exercise Set 12
*1. The CobbDouglas production function for a particular manufacturer is given by
P (x, y) = 100x3/5 y 2/5
where x represents the units of labour (at 150 per unit) and y represents the units of
capital (at 250 per unit).
Using the method
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
Exercise Set 18
1. Find the general solution of each of the equations by any appropriate method.
(a) xy
dy
y 2 = 3x2 e2y/x
dx
(b) x2
dy
+ 2xy = 6x2
dx
2. Solve the following dierential equations with the help of the given substitutions:
dy
+ y = x3 y 6
d
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
Exercise Set 16
*1. Given the inverse demand function yd = 116 + 2x x2 and inverse supply function
ys = (x + 2)2 , where x represents quantity and yd and ys each represents price, nd the
consumers surplus and the producers surplus.
2. Use partial fraction
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
Exercise Set 19
*1. Describe the behaviour as x of the general solutions of the equations in question 4
of Exercise Set 18.
2. Find the general solutions of the following dierence equations; describing their behaviour
as x
(i) yx+2 yx+1 2yx = 0
(ii)
yx+3
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
Exercise Set 17
1. Classify each of the dierential equations below in one or more of the following categories:
(i) linear equation (ii) separable equation (iii) exact equation
Find the general solutions of the equations by any appropriate method.
*(a) (x2
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
Exercise Set 9
1. The length and width of a rectangle decrease at the rate of 2 cm per minute and 3 cm per
minute respectively. When the length is 6m and the width is 3m how fast are the following
changing:
(a) the area
and
*(b) the diagonal?
2. Suppose t
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2015
Exercise Set 14
1. = Use Maple to plot the function f : R R dened by
y = x4 6x2 + 12 .
Find a formula for a local inverse to the function at each of the points x = 1, 1, 2. Sketch
the graph of each local inverse function.
Calculate the derivative of the l
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture Twenty
20. Vector Spaces, 4 of 4
In this section, we develop the concepts of angle and length, so that these concepts
can be applied to a general vector space V where angles and lengths do not arise
in a natural way. For exam
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture TwentyEight
28. Linear Transformations, 6 of 6
We now focus on symmetric matrices and a special form of diagonalisation applicable
to symmetric matrices, known as orthogonal diagonalisation. We begin by introducing orthogona
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture TwentySeven
27. Linear Transformations, 5 of 6
27.1 Diagonalisation
In the last lecture we saw examples of linear transformations T : R2 R2 where a
basis B of R2 can be found such that the matrix ABB
representing T is diagon
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture ThirtyTwo
32. Multivariate Calculus, 4 of 5
32.1 The second derivative of a function
d2 f (x)
of a function f : Rn R is dened by taking the
dx2
derivative of the (transpose of the) derivative of f (x); that is, the transpose
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Fall 2014
Mathematical Methods, Lecture ThirtySix
36. Dierential and Dierence Equations, 3 of 5
36.1 Dierence equations
Dierence equations are recurrence relations satised by a sequence cfw_yx . The
sequence cfw_yx is the unknown of the problem, and x N is an ind
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
If A = 0, what does this tell you?
Objectives
Ax = 0 = 0 x
has a nontrivial solution
What is orthogonal diagonalization?
So 0 is an eigenvalue of A
When can we do it?
How can we use it?
 A = 0
v is an eigenvector corresponding to = 0
But rst . . .
v
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
The same is true of other sets of vectors
Objectives
(1)
To nd out how to change basis.
(2)
To discover why we might want to do this.
cfw_v1 , v2 , vn Rn is a basis of Rn
Every vector x Rn can be expressed
as a unique linear combination:
1
x = a1 v1 + a
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Calculus Lecture Notes, Week Sixteen
1
Integration: Techniques and Applications
1.1
Indenite and Denite Integrals and the Fundamental
Theorem of Calculus
An indenite integral of a function f (x) is a function
that its derivative is f (x); that is,
d
dx
f
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Calculus Lecture Slides, Week Two
1
LINES, PLANES CURVES AND CONICS
1.1
Lines in R2
Given two distinct points (p1, p2) and (q1, q2) in
R2, let their position vectors be p =
q=
p1
p2
and
q1
. For a parametric equation we need
q2
a position vector on the li
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Calculus Lecture Notes, Week Seventeen
1
Separable, Exact and Linear Ordinary Dierential Equations and Introduction to Partial Differential Equations
1.1
Ordinary and Partial Dierential Equations
A dierential equation is an equation which contains at leas
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Calculus Lecture Notes, Week Eleven
1
1.1
Convex Sets, Convex and Concave Functions
and Constrained Optimisation
Complete classication of symmetric matrices
In week 10, we classied symmetric matrices by using their principal minors. We
introduced positive
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Calculus Lecture Notes, Week Fifteen
1
1.1
Interest Compounding and
Arithmetic and Geometric Sequences
Interest Compounding
Consider the following three simple problems:
Problem 1: An amount P , called the principal, is invested for t years at an annual
i
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Calculus Lecture Notes, Week Eighteen
1
Homogeneous ODEs and Higher Order Linear
ODEs with Constant Coecients
1.1
Homogeneous ODEs
Recall that a function f (x, y ) is called homogeneous of degree n if
f (x, y ) = n f (x, y ).
A homogeneous ordinary dieren
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Objectives
Last week two concepts: linear independence and span
The set cfw_v1 , v2 , , vr is a basis of a vector space V .
The set of vectors cfw_v1 , v2 , . . . , vk spans V
What does it mean?
Why do we need it?
V = Lincfw_v1 , v2 , . . . , vk = cfw_
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
If cfw_v1 , v2 , . . . , vk is a basis of V
Objectives
(1) Find the vector spaces associated to a system of equations,
then every vector in V is a unique linear combination
v = 1 v 1 + 2 v 2 + + k v k
Ax = b
Why?
dim(V ) = k
To understand the solutions
T
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
x3 2x2 2x 3 = (x 3)(x2 + x + 1)
Example.
Objectives
(1) Know what a complex number is.
x2 + x + 1 = 0
Solve
(2) Know the ways to express it,
be able to change between them,
and know when and how to use them.
1
3
Lecture 12 Complex Numbers
Fundamental Th
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
RS (A) = Lincfw_r1 , r2 , . . . , rm Rn
row space of A
Objectives
CS (A) = Lincfw_c1 , c2 , . . . , cn Rm
column space of A
(1) Subspaces associated with a matrix A
What are they?
N (A) = cfw_x  Ax = 0 Rn
null space of A
How are they related?
(2) Leadi
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Lecture 8 Linear Span Linear Independence
Objectives
(1)
Every vector x R3
cfw_v1 , v2 , , vk spans a vector space V
can be expressed as a unique linear combination:
What does it mean?
How to determine if cfw_v1 , v2 , , vk spans V ?
(2)
x
1
0
0
x = y =
University of London The London School of Economics and Political Science
Mathematical mathods
MATHEMATIC ma100

Spring 2014
Objectives
(1)
Find out about the
range and null space
Definitions.
of a linear transformation.
T : V W
null space (kernel) of T
(2)
Apply this to calculus.
range (image) of T
N ( T ) = cfw_ v V  T ( v ) = 0
To discover the linear properties of dierentia