THE BINOMIAL THEOREM
EXERCISE SET 1:
1. For all integers n 2 we have
n
n2
=
n(n 1)
.
2
2. For all integers n, k with n k + 1 1 we have
n
k+1
=
nk n
.
k+1 k
3 . For all integers n, k with n k 1 we have
n
n
n
+2
+
k1
k
k+1
n+2
.
k+1
=
4 . For all integers n
Solutions to Chapter 4 Exercises
Section 4.1
1. Let cfw_f~1 , f~2 , . . . , f~m be an orthogonal subset of Rn and let ~x Rn . Set
~x f~1 ~
~x f~2 ~
~x f~m ~
f~m+1 = ~x
f1
f2
fm .
kf~1 k2
kf~2 k2
kf~m k2
(a) Prove that f~m+1 is orthogonal to f~j , 1 j
Solutions to Chapter 3 Exercises
Disclaimer. I havent had time to properly proofread and vet these solutions. Please contact me
if you find any errors. Dr. Seyffarth.
Section 3.1
1. For each of the following transformations, either prove that it is a line
Solutions to Chapter 2 Exercises
Section 2.1
1. Define what it means for V to be a vector space, including
(a) carefully stating the axioms for vector addition;
(b) carefully stating the axioms for scalar multiplication.
Solution. See lecture notes.
2. Le
Solutions to Chapter 1 Exercises
Section 1.1
1. Prove that
r+1
U = s r, s, t R, r + t = 1
t
is a subspace of R3 .
Solution. First note that the condition r + t = 1 is equivalent to the condition that
(r + 1) + t = 0. Taking r = 1 and s = t = 0, we have
RATIONAL AND IRRATIONAL NUMBERS
DEFINITION:
Rational numbers are all numbers of the form
EXAMPLE:
p
, where p and q are integers and q = 0.
q
1
5
50
, , 2, 0, , etc.
2
3
10
NOTATIONS:
N = all natural numbers, that is, 1, 2, 3, . . .
Z = all integer number
SOLUTIONS OF HOMEWORK
PROBLEMS
2.31 If a1 , a2 , . . . , at1 , at are elements in a group G, prove that
(a1 a2 . . . at1 at )1 = a1 a11 . . . a1 a1 .
t
t
2
1
Solution:
By (v) of Theorem 3 we have
(a1 a2 . . . at1 at )1 = a1 (a1 a2 . . . at1 )1 = a1 a11 (a
Mathematical Induction
Theorem 1: Prove that
n(n + 1)
2
1 + 2 + 3 + . + n =
()
for any integer n 1.
Proof:
STEP 1: For n=1 () is true, since
1=
1(1 + 1)
.
2
STEP 2: Suppose () is true for some n = k 1, that is
1 + 2 + 3 + . + k =
k (k + 1)
.
2
STEP 3: Pro
GROUPS
DEFINITION:
A group is a set G which is equipped
with an operation and a special element e G, called the identity, such
that
(i) the associative law holds: for every
x, y, z G,
x (y z ) = (x y ) z ;
(ii) e x = x = x e for all x G;
(iii) for every x
GREATEST COMMON DIVISOR
DEFINITION:
The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor
of integers a and b.
THEOREM: If a and b are nonzero integers, then their gcd is a linear combination of a and
b, that is th
Applications of Fermats Little Theorem and
Congruences
Denition:
Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by
a b mod m,
if m | (a b).
Example:
3 1 mod 2,
6 4 mod 2,
14 0 mod 7,
25 16 mod 9,
43 27 mod 35.
Propertie
Fermats Little Theorem
Theorem (Fermats Little Theorem): Let p be a prime. Then
p | np n
(1)
for any integer n 1.
Proof: We distinguish two cases.
Case A: Let p | n, then, obviously, p | np n, and we are done.
Case B: Let
p | n.
(2)
Consider the following
Theorem 1 : We have
3 | n3 n
Proof :
()
for any integer n 1.
STEP1 : For n = 1 () is true, since
3 | 13 1.
STEP2 : Suppose () is true for some
n = k 1, that is
3 | k3 k.
STEP3 : Prove that () is true for n = k + 1 :
3 | (k + 1)3 (k + 1).
We have :
(k + 1)
GREATEST COMMON DIVISOR
DEFINITION:
The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor
of integers a and b.
THEOREM: If a and b are nonzero integers, then their gcd is a linear combination of a and
b, that is th
THREE PROBLEMS
PROBLEM 1: Prove that 2 is irrational. a Proof: Assume to the contrary that 2 is rational, that is 2 = , where a and b are integers b and b = 0. Moreover, let a and b have no common divisor > 1. Then 2= a2 b2 2b2 = a2 . (1)
Since 2b2 is ev
TWO PROBLEMS
PROBLEM 1: Prove that
2 is irrational.
Proof: Assume to the contrary that 2 is rational, that is
a
2= ,
b
where a and b are integers and b = 0. Moreover, let a and b have no common divisor > 1. Then
2=
a2
b2
2b2 = a2 .
(1)
Since 2b2 is even,
Solutions to Chapter 5 Exercises
Section 5.1
a+c
1. Define the linear transformation T : P2
by T (a + bx +
=
for all
bca
polynomials a + bx + cx2 P2 . If B = cfw_b1 , b2 , b3 is a basis of P2 and D = cfw_d~1 , d~2 is a
basis of R2 such that
1
0
b1