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
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
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 use
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
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 cal
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
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
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 th
MA212 Further Mathematical Methods
Lecture 20: Riemann-Stieltjes Integral
How to calculate them
Previously on .
For
a non-decreasing function (x) on [a, b] , and
a bounded function f (x) on [a, b] ,
w
MA212 Further Mathematical Methods
Lecture 10: Improper Integral
basic problems
toolkit functions
convergence testing
Previously on .
Substitution method (change of variable) for Riemann
Integral
Z
(b
MA212 Further Mathematical Methods
Lecture 11: Improper Integral Testing
Direct Comparison Test
Limit Comparison Test
Homework
Exercises 6:
write solutions to problems 3ab, 4, 5, 6, 7
deadline: follow
MA212 Further Mathematical Methods
Lecture 16: Manipulation of Improper Integrals
Manipulation of Improper Integrals
Basics of Laplace Transform
Previously on .
To justify the change of the order of i
MA212 Further Mathematical Methods
Lecture 2: Calculating Limits
Basic Tools
Basic Rules
Basic Methods
Taking in: + and
Organisational information
Exercises 1 are available from MA212 Moodle page;
Ha
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
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. S
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
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
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 worki
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
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 erro
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; pl
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 e
MA212 Further Mathematical Methods
Lecture 9: More on Double Integral
change of variables
Homework
Exercises 5:
write solutions to problems 1, 2, 3, 4ab
deadline: follow instructions laid down by your
MA212 Further Mathematical Methods
Lecture 19: Riemann-Stieltjes Integral
Motivation
Definition and Existence
First Example
Homework
Exercises 10:
write solutions to problems 1,3,4,5,7
deadline: follo
MA212: Further Mathematical Methods (Calculus) 201718
Exercises 5: FTC (again), transformations and more double integrals
For these and all other exercises on this course, you must show all your worki
MA212: Further Mathematical Methods (Calculus) 201718
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 Sub
MA212: Further Mathematical Methods (Calculus) 201718
Solutions to Exercises 7
1. (a) The integral
Z
0
M
1
x
dx = log(1 + M 2 ),
2
1+x
2
tends to infinity as M , which means that the integral
Z
x
dx,
MA212: Further Mathematical Methods (Calculus) 201718
Solutions to Exercises 9
1. For (a), taking the Laplace transforms of both sides of the differential equation, we get
s2 y(s) sy(0) y 0 (0) y(s) =
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 6
1. How do you sketch a graph? Do you plot the values of the function at 10 points and join the
dots? Is that what you lea
MA212: Further Mathematical Methods (Linear Algebra) 201718
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
MA212: Further Mathematical Methods (Linear Algebra) 201718
Exercises 3: Similar matrices and real inner products
For these and all other exercises on this course, you must show all your working.
1. S
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 7
1. (a) The integral
Z
0
M
1
x
dx = log(1 + M 2 ),
2
1+x
2
tends to infinity as M , which means that the integral
Z
x
dx,