MA212
CALCULUS
NOTES
Adapted from the 2015 course pack and lecture slides
Prepared by Luke Milsom
MARCH 2015
Chapter 1
Pre-integration
1.1
Limits
In this section we give an informal overview of what a limit is and what it means
to say that a function f (x
MA212: Further Mathematical Methods (Linear Algebra) 201617
Exercises 10: Generalised inverses and Fourier series
For these and all other exercises on this course, you must show all your working.
1
0
0 1
1 1
0
0
.
1. Find the strong generalised inverse of
MA212: Further Mathematical Methods (Linear Algebra) 201516
Exercises 5: Complex matrices
For these and all other exercises on this course, you must show all your working.
1.
(a) Which, if any, of the following matrices are
that tell you about their eigen
MA212: Further Mathematical Methods (Linear Algebra) 201516
Exercises 9: Least squares approximations and inverses
For these and all other exercises on this course, you must show all your working.
1.
Find an orthogonal and a non-orthogonal projection in R
MA212: Further Mathematical Methods (Linear Algebra) 201516
Solutions to Exercises 9: Least squares approximations and inverses
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1.
For t
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 10
1. This is the cumulative distribution function of a random variable X with P(X = 0) = 1/3 and
P(X = 1) = 1 1/3 = 2/3. We hope that the integral will be equal to
1
2
2
0+ 1= .
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 8
1. What you need is Leibnizs rule for differentiating an integral with respect to a parameter (see
Section 9.6 of the Subject Guide). But if you understand whats going on, its
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 9
1. Taking the Laplace transforms of both sides of the differential equation, we get
s2 f(s) sf (0) f 0 (0) f(s) =
1
,
s2
and so, using the fact that f (0) = 1 and f 0 (0) = 0,
MA212: Further Mathematical Methods (Linear Algebra) 201617
Exercises 3: Similar matrices and real inner products
For these and all other exercises on this course, you must show all your working.
1. Suppose that A and B are similar matrices. Show that A2
MA212: Further Mathematical Methods (Linear Algebra) 201617
Exercises 1: Assumed background
For these and all other exercises on this course, you must show all your working.
1.
Determine if the following matrices have inverses and,
1 1
1 1 1
1 1 0
0 0
(i)
MA212: Further Mathematical Methods (Linear Algebra) 201617
Exercises 4: Orthogonal matrices and complex inner products
For these and all other exercises on this course, you must show all your working.
1. Let u = (a, b, c)t be a vector in R3 with a, c 6=
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 10: The Riemann-Stieltjes integral
For these and all other exercises on this course, you must show all your working.
Note: The first two questions below explore the definition of the Riemann-
MA212: Further Mathematical Methods (Linear Algebra) 201617
Exercises 2: Wronskians and Linear Transformations
For these and all other exercises on this course, you must show all your working.
1.
Use the Wronskian to show that the set cfw_ex , xex , x2 ex
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 9: Laplace transforms
For these and all other exercises on this course, you must show all your working.
1.
Use Laplace transforms to solve the differential equation
f 00 (t) f (t) = t,
subjec
MA212: Further Mathematical Methods (Linear Algebra) 201516
Solutions to Exercises 1: Assumed background
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1.
The first and the third matr
MA212: Further Mathematical Methods (Linear Algebra) 201516
Solutions to Exercises 7: Dominant eigenvalues and unitary diagonalisation
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1
MA212
LINEAR ALGEBRA
NOTES
Adapted from the 2015 course pack and lecture slides
Prepared by Luke Milsom
MARCH 2015
Chapter 1
Background
1.1
Vector spaces
A vector space is a set V of vectors with two operations
1. u, v V , u + v V
2. v V and a R (for real
MA212: Further Mathematical Methods (Linear Algebra) 201516
Solutions to Exercises 3: Similar matrices and real inner products
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1.
Suppos
MA212: Further Mathematical Methods (Linear Algebra) 201516
Solutions to Exercises 6: Jordan normal forms
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1.
(a) For each t, if we take
MA212: Further Mathematical Methods (Linear Algebra) 201516
Solutions to Exercises 8: Singular values and projections
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1.
We have AA =
2
1
MA212 Further Mathematical Methods
Part II: Linear Algebra
In this pack you will find a set of notes and exercises for the second half of the course
MA212. These materials are identical to those used for the half-unit course MA201,
which has now been di
MA212: Further Mathematical Methods (Linear Algebra) 201516
Solutions to Exercises 2: Wronskians and Linear Transformations
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1.
We calcul
MA212: Further Mathematical Methods (Linear Algebra) 201516
Solutions to Exercises 4: Orthogonal matrices and complex inner products
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1.
MA212: Further Mathematical Methods (Linear Algebra) 201516
Exercises 7: Dominant eigenvalues and unitary diagonalisation
For these and all other exercises on this course, you must show all your working.
1.
You are given that M = P JP 1 where
0 2 2
0 1 1
MA212: Further Mathematical Methods (Linear Algebra) 201516
Exercises 8: Singular values and projections
For these and all other exercises on this course, you must show all your working.
1 1 0
1. Find the singular values of the matrix A =
.
1 0 1
What is
MA212: Further Mathematical Methods (Linear Algebra) 201516
Solutions to Exercises 5: Complex matrices
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1.
(a) By inspection, A and B are
MA212: Further Mathematical Methods (Linear Algebra) 201516
Exercises 3: Similar matrices and real inner products
For these and all other exercises on this course, you must show all your working.
1. Suppose that A and B are similar matrices. Show that A2
MA212: Further Mathematical Methods (Linear Algebra) 201516
Exercises 6: Jordan normal forms
For these and all other exercises on this course, you must show all your working.
1.
An n n matrix A is called nilpotent if there exists a positive integer k such
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 8: Leibnizs rule, the manipulation of improper integrals and LTs
For these and all other exercises on this course, you must show all your working.
1.
For x > 0, a function, f , is defined by
Further Mathematical Methods (Linear Algebra)
Solutions For The 2002 Examination
Question 1
(a) To be an inner product on the real vector space V , a function hx, yi which maps vectors x, y V
to R must be such that:
i. Positivity: hx, xi 0 and, hx, xi = 0