Lecture 17
Complex exponential, de Moivres theorem.
De Moivres theorem
[cos t + i sin t]n = cos nt + i sin nt
Since eit = cos t + i sin t, this is the same as
it n = eint
e
Proven by induction.
I.17.1
Example 17.1 Find (1 + i)5
I.17.2
We use de Moivres
Lecture 13
Proofs
In mathematics there are rigorous rules of deduction needed to
prove a fact.
We must state denitions and assumptions, which often requires
translating sentences to mathematical language.
Example 13.1 Prove that the square of every non-ze
Lecture 5
Matrix inverses, augmented matrices, matrix equations
Recall that the inverse of a square matrix A is a matrix A 1
such that AA 1 = I.
Example 5.1
2
1
6
A=4 2
2
3
0 1
7
1 3 5
2 1
)A 1=
How do we nd A 1?
The sequence of row operations that reduce
Lecture 20
Subspaces
Certain subsets, S, of Rm have the very special property that
adding vectors in S and multiplying vectors by scalars never
takes you to a vector that is not also in S.
These special sets are called subspaces of Rm.
Example 20.1 The x-
Lecture 28
Angles, Cauchy-Schwarz inequality.
The Cauchy-Schwarz inequality
For an inner product h, i on V dene the angle between two
vectors u, v 2 V by
This requires that
Theorem 28.1
hu, vi
cos =
.
kukkvk
hu, vi
1
1.
kukkvk
Let u and v be vectors in a
Lecture 24
Solution spaces, row and column spaces.
Finding Bases The method depends on the situation.
Solution of a System of Homogeneous Equations
Example 24.1 Find a basis (for the subspace)
V = cfw_(x1, x2, x3, x4) : x1 + 2x2 + x3 + x4 = 0,
3x1 + 6x2 +
Lecture 32
IMAGE AND KERNEL OF T ;
RANK AND NULLITY OF T
Why linear? Recall that a line can always be dened by a
parametric equation r = a + b where varies over R
If T : R2 ! R2 is linear, then
Tr =
So T transforms the line to another line through T a wit
Lecture 36
EIGENVALUES AND EIGENVECTORS, CHARACTERISTIC POLYNOMIAL
This topic builds on our work on linear transformations, especially those of the form
T :V !V.
Why are we interested in eigenthings?
There are many applications.
I.36.1
The Idea! We are i
Lecture 44
Tangent plane, linear approximation
Geometrical interpretation of @f and @f .
@x
@y
Intersect the surface by the vertical plane y = y0, to give a
@f
curve of intersection. Then
(x0, y0) gives the slope of the
@x
tangent to this curve at the poi
Lecture 40
Markov Chains
A Markov chain is a system that can occupy a number of states,
and in time the system can change with probability depending
only on the state of the system at the preceding time.
Any matrix with columns consisting of non-negative
Lecture 9
Cross product, triple products and applications.
Two non-zero vectors u and v are parallel if u = v for a scalar
2 R. They are perpendicular, or orthogonal, if u w = 0,
hence the angle between them is /2.
The vector projection of u onto v is
uv
Topic I:
Matrices and linear equations
Lecture 1
Systems of linear equations
A linear system is a set of simultaneous equations in which
each variable occurs to the power one only. The equation
2x + y = 5
describes a line. It has solution set
x = t,
y=5
2
Mathematics 620-157
Typical exam questions
Typical Exam Questions 1
1. (a) Find all solutions, z, of the equation
(z 2 4i)(z 2 + 4z + 20) = 0 ,
expressing your answers in cartesian form.
(b) Using the complex exponential, express sin3 (2) in terms of sine
1.1
Linear equation:
Homogeneous linear equation:
Consistent linear system has at least one solution (one or infinitely many). Inconsistent linear
system: has no solutions
One way to describe the solution set is to solve the equation for x in terms of y t
Lecture 9
Cross product, triple products and applications.
Two non-zero vectors u and v are parallel if u = v for a scalar
2 R. They are perpendicular, or orthogonal, if u w = 0,
hence the angle between them is /2.
The vector projection of u onto v is
uv
MAST10008 Assignment 3: Due noon 14th April in your tutors assignment box.
(1) Prove carefully that the number of primes factors of an integer n > 1 is strictly less than n.
(2) Prove that 5 3 is irrational.
7
(3) In what position does the number
appear i