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.
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 langu
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 d
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 speci
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 =
.
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
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 =
S
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?
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) giv
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 ti
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 betwe
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
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
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
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 solu
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 betwe
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 irra