Prelims Linear Algebra I
Michaelmas Term 2014
1
Systems of linear equations and matrices
Let m, n be positive integers. An m n matrix is a rectangular array, with nm numbers,
arranged in m rows and n columns. For example
1 5 0
3 0 2
is a 2 3 matrix. We al
A. Algebra 1 - Linear Algebra (1)
Please read sections 2.6, 9.1 - 9.4, 11.1 and 11.2 from Kaye and Wilsons Linear
Algebra. Work on the following problems.
1. (a) Prove that Fp = cfw_0, 1, . . . , p 1, the set of equivalence classes of integers modulo a pr
A. Algebra 1 - Linear Algebra (5)
You may like to read Sections 13, 15, 17, 20 from Finite Dimensional Vector Spaces
by P. Halmos and/or chapter 11 from Linear Algebra by Lipschutz and Lipson.
1. For each of the basis of R3 nd the dual basis:
(i) cfw_(1,
A. Algebra 1 - Linear Algebra (6)
Please read sections 5.1 - 5.4 from Kaye and Wilsons Linear Algebra.
1. (Kaye & Wilson, Exercise 5.1(c) Use the Gram-Schmidt process to obtain
an orthogonal basis for V , the vector space of polynomials of degree less or
A. Algebra 1 - Linear Algebra (3)
Please read sections 12.1 12.3 from Kaye and Wilsons Linear Algebra.
1. (Kaye & Wilson, Exercise 12.4) (a) Find two 2 2 matrices over R which
have the same characteristic polynomial but which are not similar.
(b) Find two
A. Algebra 1 - Linear Algebra (7)
Please read sections 13.1 -13.4 from Kaye and Wilsons Linear Algebra and/or
chapter 13 from Lipschitz and Lipsons Linear Algebra.
1. Let V be an inner product space and v V .
(a.) Suppose T is self-adjoint. Show that T 2
A. Algebra 1 - Linear Algebra (8)
Linear Algebra is one of the backbones of modern mathematics. The enclosed short
projects are intended to give a glimpse of a variety of further topics and applications.
They are not much more than extended problems but h
1 . V E C T O R S PA C E S , L I N E A R M A P S , A N D
RINGS
1
Recall some standard theory on vector spaces.
Denition 1.1: A set F with two binary operations + and is a
eld if both (F, +, 0) and (F \ cfw_0, , 1) are abelian groups and
the distribution l
3 . Q U O T I E N T S PA C E S
1
Let V be a vector space over a eld F and let U be a subspace.
Denition and Lemma 3.1: The set of cosets V/U = cfw_v + U |
v V with the operations
(v + U ) + (w + U ) := v + w + U
a(v + U ) := av + U
for v, w V and a F is
A. Algebra 1 - Linear Algebra (4)
Please read sections 14.1 14.4 from Kaye and Wilsons Linear Algebra.
1. Let T : V V be a linear transformation and suppose that for some v V ,
T k (v) = 0 but T k1 (v) = 0. Prove that the set B = cfw_T k1 (v), . . . , T (
A. Algebra 1 - Linear Algebra (2)
Please read sections 8.1 8.3, 10.1 and 10.2 (mainly for revision), as well as 10.3
and 11.3 from Kaye and Wilsons Linear Algebra.
1. Suppose U is a subspace of V invariant under a linear transformation T :
V V . Prove tha
Prelims Linear Algebra, MT 2015
Sheet 2 (for tutorials in week 4; * = optional question)
Qu. 1
Find the reduced
1 2
B = 1 1
3 6
row echelon forms of the matrices B and
1 2 3 2
3 1 2
4 1 3 ,
C = 2 5 8 1
1 4 7 4
13 3 11
C where
4 1
6 2 ,
0
where R.
Qu. 2 F
Mathematics Prelims: Linear Algebra, MT 2015
Sheet 1 (for tutorials in week 3; * = optional question)
Qu. 1 We say that square matrices A and B of the same size commute with respect
to multiplication if AB = BA. Now let A be a 2 2 matrix with entries in R
Prelims Linear Algebra, MT 2015
Sheet 3 (for tutorials in week 5; * = optional question)
Qu. 1 Let L, M, N be subspaces of a vector space V .
(a) Prove that
(L M ) + (L N ) L (M + N )
Give an example of subspaces L, M, N of R2 where (L M ) + (L N ) = L (M
Prelims Linear Algebra, MT 2015
Sheet 4 (for tutorials in week 6; * = optional question)
Qu. 1 (a) Which of the following sets of vectors in V = R3 is a basis for V ?
(i) cfw_(1, 3, 0), (2, 3, 4), (3, 0, 4),
(ii) cfw_(1, 2, 3), (2, 3, 1), (3, 1, 2).
(b) L
Prelims Linear Algebra, MT 2015
Sheet 5 (for tutorials in week 7; * = optional question)
Qu. 1
Let V = R5 , and dene subspaces U1 and U2 of V by
U1 = cfw_(x1 , . . . , x5 ) V : x1 + x2 = 0,
5
U2 = cfw_(x1 , . . . , x5 ) V :
5
(1)i xi = 0.
xi = 0 and
i=1
i
Prelims Linear Algebra, MT 2015
Sheet 6 (for tutorials in week 8; * = optional question)
Qu. 1 Let U, V, W be vector spaces over R. Assume that T and T are linear transformations from U to V , and S is a linear transformation from V to W .
(a) Check that
Prelims Linear Algebra, MT 2015
Sheet 7 (for the vacation; * = optional question)
Qu. 1
Find the inverse of the following matrix using elementary row operations:
1 1
0
0
1
0 1
0
.
A=
1
0
0 1
0
1
1
1
Qu. 2 Let T : R3 R3 be the linear map
T (x, y, z) =
Linear Algebra II
Alan Lauder
February 12, 2015
1
Determinants
In this course we study in greater depth linear maps and show that often one can nd bases so that
the matrices for such maps take a particularly nice form. In Section 3.2 a geometric applicati
Prelims Linear Algebra, MT 2015
Sheet 3 (for tutorials in week 5; * = optional question)
Qu. 1 Let L, M, N be subspaces of a vector space V .
(a) Prove that if the sum L + M of subspaces L, M is dened as in Sheet 2 Qu 6 then
(L M ) + (L N ) L (M + N )
Giv
2 . P O LY N O M I A L S
1
We will discuss polynomials over a eld F in more detail.
Theorem 2.1: Division algorithm for polynomials
Let f ( x ), g( x ) F[ x ] be two polynomials with g( x ) = 0. Then
there exists q( x ), r ( x ) F[ x ] such that
f ( x ) =
4. TRIANGULAR FORM AND THE
C AY L E Y- H A M I LT O N T H E O R E M
1
Let T : V V be a linear transformation. A subspace U V
is called T-invariant if T (U ) U. Let S : V V be another
linear map.
Lemma 4.1: If U is T- and S-invariant, then U is also invari
5 . T H E P R I M A RY D E C O M P O S I T I O N T H E O R E M
1
We will need the following consequence of the Division Algorithm.
Proposition 5.1: Let a, b F[ x ] be non-zero polynomials and
assume ( a, b) = c. Then there exist s, t F[ x ] such that:
a(
BT11803 MATHEMATICAL ECONOMICS
Tutorial 3
1.
A salesman has a monthly income given by
, where S is the
number of products sold in a month. How many products must he sell to make
at least RM4500 a month?
2.
The ABX company management would like to know the
BT11803 MATHEMATICAL ECONOMICS
Tutorial 8
1.
Find the following limit for ( )
(
)
( )
:
.
( )
2.
The revenue function for a certain product is given by
Find
( ).
3.
The cost of a raw material increases according to the equation ( )
. The rate at which the
BT11803 MATHEMATICAL ECONOMICS
Tutorial 5
1.
Suppose RM2000 is invested at 13% compounded annually. Find the value of
the investment after five years. Find the interest earned over the first five
years.
(compound interest formula)
Therefore the value of t
BT11803 MATHEMATICAL ECONOMICS
Tutorial 2 - Answers
1.
Suppose the weekly demand function for large pizzas at local pizza parlor is
a.
If the current price is RM18.50 per pizza, how many pizzas are sold each
week?
(
)
b.
If 200 pizzas are sold each week,
BT11803 MATHEMATICAL ECONOMICS
Tutorial 4 - Answers
1.
A clerk purchased a new car in 2001 for RM32,000. In 2004, he sold it to a
friend for RM26,000. Draw a line showing the relationship between the selling
price of the car and the year in which it was s
BT11803 MATHEMATICAL ECONOMICS
Tutorial 6
1.
A manufacturer of doors, windows, and cabinets writes her yearly profit (in
thousands of RM) for each category in a column vector as
[
[
]. Her fixed costs of production can be described by the vector
]. She ca
BT11803 MATHEMATICAL ECONOMICS
Tutorial 6
1.
A manufacturer of doors, windows, and cabinets writes her yearly profit (in
thousands of RM) for each category in a column vector as
[
]. Her fixed costs of production can be described by the vector
[
]. She ca