MAS367/MAS467/MAS6003 Linear and Generalised Linear Models
1
Linear and Generalised Linear Models
Exercise Sheet 2
Submit only those questions marked with (*)
Submit your work either in the class or by email to [email protected] The deadline
is Thur
INVERSE FUNCTION RULE AND CURVE SKETCHING
5 minute review.
Briey cover the inverse trigonometric functions and their
1
derivatives (and remind students that that sin1 x does not mean sin x ).
d
1
Class warm-up. Run through the proof that dx (cos1 x) = 1x2
INVERSE FUNCTIONS, EXPONENTIALS AND LOGARITHMS
5 minute review.
things like
A brief reminder of inverse functions, exp and ln covering
f 1 (f (x) = x and f (f 1 (x) = x;
y = f (x) and y = f 1 (x) have graphs which are reections in y = x;
range(f ) = do
EULERS RELATION
5 minute review.
Review Eulers relation, ei = cos + i sin , commenting
briey on how it follows from the Maclaurin series of exp, sin and cos. Also cover
the exponential forms of sin and cos, namely
sin =
ei ei
2i
and
cos =
ei + ei
.
2
Clas
GENERAL LOGARITHMS AND HYPERBOLIC FUNCTIONS
5 minute review. Remind students
what loga x is for general a > 1 (i.e., loga x is the power of a needed to make
x) and that ln = loge ;
the denitions of sinh, cosh and tanh;
the identities cosh2 (x) sinh2 (x
DE MOIVRES THEOREM
5 minute review. Recap de Moivres Theorem, cos(n) + i sin(n) = (cos +
i sin )n , and how to solve z n = r(cos +i sin ) for z, perhaps by doing the warm-up
below.
Class warm-up. Find the cube roots of 1 + i.
Problems. Choose from the bel
CURVE SKETCHING
5 minute review. Using the graph of y =
as an example, remind students briey
1
x1
(or any other suitably easy curve)
what a graph is;
the dierence between plotting (by calculating) and sketching (by reasoning);
the importance of labelli
COMPLEX NUMBERS
5 minute review. Remind students what a complex number is, how to add and
multiply them, what the complex conjugate is, and how to do division. They will
also need to know that a root of a polynomial f (x) is a solution to f (x) = 0, and
t
DIFFERENTIATION RULES
5 minute review. Remind the students of the addition, product, quotient and
dy
chain rules, the latter as both if y = f (g(x) then dx = g (x)f (g(x) and if
dy
dy
y = f (u) with u = u(x), then dx = du . du .
dx
Class warm-up. Find the
DIFFERENTIATION 1: FIRST RESULTS
df
5 minute review. Recall the denition dx (x0 ) = limh0 f (x0 +h)f (x0 ) by working
h
out the gradient of the line on the curve between (x0 , f (x0 ) and (x0 + h, f (x0 + h).
Perhaps do an example, such as y = x3 at x0 =
THE ARGAND DIAGRAM
5 minute review. Review the Argand diagram (aka Argand plane, aka complex
plane), the modulus and argument and the polar form of a complex number, z =
r(cos + i sin ). Also cover the eect of addition and multiplication.
Class warm-up. P
MAS439: Problem Sheet 3
1. Let R be a commutative ring, and let U and V be multiplicative subsets
of R.
(i) Show that U V is a multiplicative subset of R.
Solution We must show that 1 U V , and that U V is closed
under multiplication. Since 1 U and 1 V we
MAS439: Problem Sheet 5
1. Check Nakayamas lemma (Version 1) by hand where we take our ring R
to be Z, our module M to be Z/10Z, and our ideal I to be (7).
In other words, show that IM = M , and exhibit any element x 1 + (7)
with xM = 0.
(3 marks)
Solutio
MAS439: Problem Sheet 1
1. Let R be the ring R[t]/(t2 1).
(i) Let V be a vector space over R, and let T be a linear involution (that
is, a linear map V V such that T 2 is the identity on V ).
Show that there is an R-module structure on V for which tv = T
MAS439: Problem Sheet 4
1. Whats the Jacobson radical of the following rings:
(a) Z?
(b) the localisation Z(11) ?
(c) the ring Z/20Z?
(3 marks)
Solution
(a) The maximal ideals of Z are (p) where p is prime. There are no
elements in common between all thes
MAS439: Problem Sheet 2
1. Let R be the ring Z/4Z of integers modulo 4.
(i) Show that there is exactly one R-module, N , with two elements.
Solution Let the elements be cfw_0, a. We must have 2a = a + a = 0.
Hence we have 3a = a + a + a = a, and this desc
MAS439: Problem Sheet 7
1. The real numbers 3 and 7 are both integral over Z, being roots respectively of x2 3 and x2 7. Its a consequence of Corollary 13.2 that their
sum 3 + 7 is also integral over Z.
Prove this directly, by exhibiting a monic polynom
MAS439: Problem Sheet 6
1. Let Z3 Z3 be the map of Z-modules dened by the matrix
2 4
4 0
0 2
2
2 .
6
Find a monic polynomial equation satised by with all coecients (except the leading coecient) divisible by 2.
(3 marks)
Solution
We compute
2
det 4
0
4
2
MAS439: Problem Sheet 8
Due: Friday, 2nd May, 4pm
1. The eld R is not algebraically closed. Give a maximal ideal I of R[x]
which is not of the form (x a) for any a R. Prove that the ideal is
maximal by identifying the quotient eld R[x]/I as a well-known e
MAS439: Problem Sheet 7
Due: Friday, 4th April, 4pm
(Corrected slightly on 1st April)
1. The real numbers 3 and 7 are both integral over Z, being roots respectively of x2 3 and x2 7. Its a consequence of Corollary 13.2 that their
sum 3 + 7 is also integ
MAS439: Problem Sheet 2
Due: Friday, 28th February, 4pm
1. Let R be the ring Z/4Z of integers modulo 4.
(i) Show that there is exactly one R-module, N , with two elements.
(ii) Show that there are exactly two dierent R-modules, M1 and M2 ,
with four eleme
MAS439: Problem Sheet 1
Due: Friday, 21st February, 4pm
1. Let R be the ring R[t]/(t2 1).
(i) Let V be a vector space over R, and let T be a linear involution (that
is, a linear map V V such that T 2 is the identity on V ).
Show that there is an R-module
MAS439: Problem Sheet 4
Due: Friday, 14th March, 4pm
1. Whats the Jacobson radical of the following rings:
(a) Z?
(b) the localisation Z(11) ?
(c) the ring Z/20Z?
(3 marks)
2. Let U Z be the set of integers which are not multiples of 3 or 7; in other
word
MAS439: Problem Sheet 3
Due: Friday, 7th March, 4pm
1. Let R be a commutative ring, and let U and V be multiplicative subsets
of R.
(i) Show that U V is a multiplicative subset of R.
(ii) Show that the set cfw_uv u U, v V is a multiplicative subset of R.
MAS439: Problem Sheet 9
Due: Friday, 9th May, 4pm
1. Projectivise the following varieties, and nd all the points at innity:
(a) V (J1 ), where J1 = (xy) is an ideal in C[x, y];
(b) V (J2 ), where J2 = (x2 + y 2 1) is an ideal in R[x, y];
(c) V (J3 ), wher
MAS439: Problem Sheet 5
Due: Friday, 21st March, 4pm
1. Check Nakayamas lemma (Version 1) by hand where we take our ring R
to be Z, our module M to be Z/10Z, and our ideal I to be (7).
In other words, show that IM = M , and exhibit any element x 1 + (7)
w
MAS439: Problem Sheet 6
Due: (Unusually!) Tuesday, April 1st, 4pm
1. Let Z3 Z3 be the map of Z-modules dened by the matrix
2 4
4 0
0 2
2
2 .
6
Find a monic polynomial equation satised by with all coecients (except the leading coecient) divisible by 2.
(3
MAS114 Solutions
Dr James Cranch
Week 2
1. There are many examples: we just give a few possibilities.
(a) Examples include 1 and 123456789;
(b) Examples include 1cfw_2 and 5cfw_7;
?
(c) Examples include , 2, and tanp30q.
2. The sets are:
(a) t0, 1, 2u;
(b
MAS114 Exercises
Dr James Cranch
Week 2
1. Give an example (without proof) of:
(a) a number which is in Z but not in N;
(b) a number which is in Q but not in Z;
(c) a number which is in R but not in Q.
2. Write down all the elements of each of the followi
MAS114 Exercises
Dr James Cranch
Week 1
1. Learn the Greek alphabet: learn the names of all the lower-case letters
, , , , , , , , , , , , , , o, , , , , , , , , .
(Theyre among the commonest of the unfamiliar symbols that mathematicians use.)
2. Work out