University of London The London School of Economics and Political Science
MA100
MA 100

Fall 2015
19 Vector Spaces, 3 of 4
19.1 Theorems on linear span and linear independence
In this subsection, we collect four theorems relating to the linear span and the linear
independence or dependence of a set of vectors cfw_v1 , v2 , . . . , vk in a vector spac
University of London The London School of Economics and Political Science
MA100
MA 100

Fall 2015
18 Vector Spaces, 2 of 4
18.1 Linear span
Recall that by a linear combination of vectors v1 , v2 , . . . , vk we mean a vector v of
the form
v = 1 v1 + 2 v2 + + k vk ,
for some constants i R.
Now suppose that V is a vector space and that the vectors v1 ,
University of London The London School of Economics and Political Science
MA100
MA 100

Fall 2015
17 Vector Spaces, 1 of 4
17.1 Definition of a vector space
A real vector space V is a nonempty set equipped with a vector addition operation
and a scalar multiplication operation such that for all , R and all u, v, w V :
1. u + v V
(closure under additio
University of London The London School of Economics and Political Science
MA100
MA 100

Fall 2015
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 example, in a vector space of matrices, wh
University of London The London School of Economics and Political Science
MA100
MA 100

Fall 2015
Exercises and Solutions from Lecture 17
Exercise 17.3.1 Consider the set V = R2 and define vector addition and scalar
multiplication as follows:
For u V and v V , vector addition is defined by numerical addition in the
standard way:
u1
v1
u1 + v1
u+v =
University of London The London School of Economics and Political Science
MA100
MA 100

Fall 2015
Week 9 Inclass Practice, Lecture 16 Material
Consider the system of linear equations Ax = b given below, where and are
constants, and
1 2 0
A = 5 1 ,
1 1 1
x
x = y ,
z
2
b = 7 .
(a) By performing row reductions on the augmented matrix (Ab), determin
University of London The London School of Economics and Political Science
MA100
MA 100

Fall 2015
Exercises and Solutions from Lecture 20
Exercise 20.7.1
follows1
An inner product h , i is defined on the Euclidean space R2 as
T
hu, vi = u
1 2
v.
2 5
(a) Show that bilinearity, symmetry and positivity are satisfied by h , i.
1
(b) Find a vector y orth
University of London The London School of Economics and Political Science
MA100
MA 100

Fall 2015
Week 9 Homework, Lecture 15 Material
Let A be an m n matrix and x be an n 1 vector of unknowns. It is known that
the reduced row echelon form of A has 3 leading ones.
(a) Can m or n be strictly less than 3? Briefly justify your answer.
[2 marks]
(b) If m
University of London The London School of Economics and Political Science
MA100
MA 100

Spring 2017
Exercises and Solutions from Lecture 28
1 2
Exercise 28.4.1 (a) Orthogonally diagonalise the matrix A =
.
2 1
(b) Use the result in part (a) to sketch the curve xT Ax = 3 in the xyplane.
Solution to Exercise 28.4.1 We solve the characteristic polynomial
University of London The London School of Economics and Political Science
MA100
MA 100

Spring 2017
LT Week 3 Inclass Practice, Lecture 24 Material
2
2
Consider the linear transformation
T : R R defined by its effect on the standard
1
0
basis B = cfw_e1 , e2 =
,
of R2 :
0
1
T (e2 ) = 2e1 e2 .
T (e1 ) = 3e1 + 2e2
(a) Find the matrix AT representing th
University of London The London School of Economics and Political Science
MA100
MA 100

Spring 2017
Exercises and Solutions from Lecture 6
Exercise 6.5.1 Consider the function f : I R given by
f (x) = x3 3x.
Find if global extrema exist and where they occur for each of the following cases:
(i) I = (, 0), (ii) I = (3, 4], (iii) I = [2, 1].
Solution to Ex
University of London The London School of Economics and Political Science
MA100
MA 100

Spring 2017
LT Week 3 Homework, Lecture 23 Material
(a) Give an example of a matrix A such that the linear transformation T : R3 R4
defined by T (x) = Ax has the following properties:
2
1
1 , 0
ker(T ) = Lin
2
3
and
0
3
R(T ) = Lin
4 .
5
[5 marks]
(b) Give
University of London The London School of Economics and Political Science
MA100
MA 100

Spring 2017
Exercises and Solutions from Lecture 27
2 1 2
Exercise 27.3.1 Consider the matrix A = 0 1 2.
0 0 3
(a) Find the eigenvalues and the corresponding eigenvectors of A.
(b) Hence, find an invertible matrix P and a diagonal matrix D such that P1 AP = D.
Soluti
University of London The London School of Economics and Political Science
MA100
MA 212

Fall 2015
MA212 Further Mathematical Methods
Lecture 4: Limits unleashed
Continuity and approximation
Taylors Theorem and its use
LHspitals Rule
Previously on .
A function f (t) is continuous at point t = c if
lim f (t) = f (c).
tc
This means: For every > 0 there e
University of London The London School of Economics and Political Science
MA100
MA 212

Fall 2015
MA212 Further Mathematical Methods
Lecture 8: Double Integral (continued)
using Fubini Theorem
volume under the curve over bounded region
Previously on .
Double Integral: We would like to
calculate the volume under the surface z = f (x, y) over a region R
University of London The London School of Economics and Political Science
MA100
MA 212

Fall 2015
MA212 Further Mathematical Methods
Lecture 3: Continuity, differentiability with
applications
Limits at a point
Continuity
Taylors Theorem
Homework
Exercises 2:
write solutions to problems 2, 3, 4a, 6, 8b, 9a
deadline: follow instructions laid down by you
University of London The London School of Economics and Political Science
MA100
MA 212

Fall 2015
MA212 Further Mathematical Methods
Lecture 6: Riemann Integral
Definition of integral
Worked example
Previously on.
LHospital Rule
Let c can be any real number or + or .
If lim f (t) = lim g(t) = 0 ,
tc
tc
or
if lim f (t) = + or , and lim g(t) = + or ,
tc
University of London The London School of Economics and Political Science
MA100
MA 212

Fall 2015
MA212 Further Mathematical Methods
Lecture 2: Calculating Limits
Basic Tools
Basic Rules
Basic Methods
Taking in: + and
Organisational information
The video recordings of lectures are not yet available;
Exercises 1 are available from MA212 Moodle page;
H
University of London The London School of Economics and Political Science
MA100
MA 212

Fall 2015
MA212 Further Mathematical Methods
Lecture 5: More on LHospitals
Rule and
Taylor Theorem
Quick recap: Taylor Theorem
More on LHspitals Rule
Taylor Series and Approximation
Homework
Exercises 3:
write solutions to problems 1ab, 3, 5, 6
deadline: follow ins
University of London The London School of Economics and Political Science
MA100
MA 212

Fall 2015
Welcome to MA 212
Further Mathematical Methods
Jozef Skokan
Calculus
Columbia House COL.3.04
Adam Ostaszewski
Linear Algebra
Columbia House COL.4.06
Department of Mathematics
London School of Economics and Political Science
Lectures
Calculus: weeks 110 in