Exercise for the break Determine the
greatest common divisor
of
X
2
−
1
{\displaystyle {}X^{2}-1}
X
3
−
1
{\displaystyle {}X^{3}-1}
Exercises Determine the
greatest common divisor
of
X
−
1
{\displaystyle {}X-1}
X
−
2
{\displaystyle {}X-2}
Determine the
greatest common divisor
of
X
4
−
X
3
+
7
X
2
−
5
X
+
11
{\displaystyle {}X^{4}-X^{3}+7X^{2}-5X+11}
X
3
−
6
X
2
+
4
X
+
6
{\displaystyle {}X^{3}-6X^{2}+4X+6}
Determine in
Q
[
X
]
{\displaystyle {}\mathbb {Q} [X]}
greatest common divisor
of the polynomials
P
=
X
3
+
2
X
2
+
5
X
+
2
{\displaystyle {}P=X^{3}+2X^{2}+5X+2}
Q
=
X
2
+
4
X
−
3
{\displaystyle {}Q=X^{2}+4X-3}
Determine in
Q
[
X
]
{\displaystyle {}\mathbb {Q} [X]}
greatest common divisor
of the polynomials
P
=
X
9
−
1
{\displaystyle {}P=X^{9}-1}
Q
=
X
3
−
1
{\displaystyle {}Q=X^{3}-1}
Determine in
R
[
X
]
{\displaystyle {}\mathbb {R} [X]}
greatest common divisor
of the polynomials
P
=
X
3
+
π
X
2
+
7
{\displaystyle {}P=X^{3}+\pi X^{2}+7}
Q
=
X
2
+
7
X
−
2
{\displaystyle {}Q=X^{2}+{\sqrt {7}}X-{\sqrt {2}}}
Determine in
F
5
[
X
]
{\displaystyle {}{\mathbb {F} }_{5}[X]}
greatest common divisor
of the polynomials
P
=
X
4
+
3
X
3
+
X
2
+
4
X
+
2
{\displaystyle {}P=X^{4}+3X^{3}+X^{2}+4X+2}
Q
=
2
X
3
+
4
X
2
+
X
+
3
{\displaystyle {}Q=2X^{3}+4X^{2}+X+3}
Let
n
≥
2
{\displaystyle {}n\geq 2}
Perform in
Q
[
X
]
{\displaystyle {}\mathbb {Q} [X]}
P
{\displaystyle {}P}
Q
{\displaystyle {}Q}
P
=
X
n
−
1
+
X
n
−
2
+
⋯
+
X
2
+
X
+
1
{\displaystyle {}P=X^{n-1}+X^{n-2}+\cdots +X^{2}+X+1}
T
=
X
−
1
{\displaystyle {}T=X-1}
Find a representation of
1
{\displaystyle {}1}
Let
K
{\displaystyle {}K}
field , and let
K
[
X
]
{\displaystyle {}K[X]}
polynomial ring
over
K
{\displaystyle {}K}
F
,
G
∈
K
[
X
]
{\displaystyle {}F,G\in K[X]}
K
⊆
L
{\displaystyle {}K\subseteq L}
F
{\displaystyle {}F}
G
{\displaystyle {}G}
K
[
X
]
{\displaystyle {}K[X]}
F
{\displaystyle {}F}
G
{\displaystyle {}G}
L
[
X
]
{\displaystyle {}L[X]}
Prove the Lemma of Bezout for integers
a
1
,
…
,
a
n
{\displaystyle {}a_{1},\ldots ,a_{n}}
The delivery company "bucket without bugs“ is specialized in the transport of water. It has three buckets, one with
77
{\displaystyle {}77}
91
{\displaystyle {}91}
143
{\displaystyle {}143}
Determine in
Z
{\displaystyle {}\mathbb {Z} }
1071
{\displaystyle {}1071}
1029
{\displaystyle {}1029}
Determine in
Z
{\displaystyle {}\mathbb {Z} }
2956
{\displaystyle {}2956}
2444
{\displaystyle {}2444}
Determine in
Z
{\displaystyle {}\mathbb {Z} }
3146
{\displaystyle {}3146}
1515
{\displaystyle {}1515}
Rabbits are born in the middle of the month, the times of gestation is one month, and they reach sexual maturity at the age of two months. Every litter consists of two rabbits, and they live forever.
We start in month
1
{\displaystyle {}1}
f
n
{\displaystyle {}f_{n}}
n
{\displaystyle {}n}
f
1
=
1
{\displaystyle {}f_{1}=1}
f
2
=
1
{\displaystyle {}f_{2}=1}
f
n
+
2
=
f
n
+
1
+
f
n
.
{\displaystyle {}f_{n+2}=f_{n+1}+f_{n}\,.}
This sequence is called the sequence of the Fibonacci numbers . How many of the
f
n
{\displaystyle {}f_{n}}
n
{\displaystyle {}n}
The Fibonacci-numbers start with
1
,
1
,
2
,
3
,
5
,
8
,
13
,
21
,
34
,
…
{\displaystyle {}1,1,2,3,5,8,13,21,34,\ldots }
Apply to two consecutive
Fibonacci numbers
the
Euclidean algorithm.
What can you observe?
Determine the
kernels
of the powers
M
i
{\displaystyle {}M^{i}}
M
=
(
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
)
.
{\displaystyle {}M={\begin{pmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{pmatrix}}\,.}
Determine the kernels of the powers
M
i
{\displaystyle {}M^{i}}
M
=
(
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
)
.
{\displaystyle {}M={\begin{pmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&0\\0&0&0&0&1\\0&0&0&0&0\end{pmatrix}}\,.}
Let
M
=
(
3
1
7
0
3
5
0
0
3
)
.
{\displaystyle {}M={\begin{pmatrix}3&1&7\\0&3&5\\0&0&3\end{pmatrix}}\,.}
Determine the
kernels
of the powers
(
3
E
3
−
M
)
i
.
{\displaystyle {\left(3E_{3}-M\right)}^{i}.}
Determine the
generalized eigenspaces
of the matrix
(
1
1
0
0
0
1
0
0
0
0
2
3
0
0
0
2
)
.
{\displaystyle {\begin{pmatrix}1&1&0&0\\0&1&0&0\\0&0&2&3\\0&0&0&2\end{pmatrix}}.}
Let
π
{\displaystyle {}\pi }
cycle
of length
n
{\displaystyle {}n}
M
{\displaystyle {}M}
permutation matrix ,
that is,
M
=
(
0
0
0
…
1
1
0
0
…
0
0
1
0
…
0
⋮
⋱
⋱
⋱
⋮
0
0
…
1
0
)
.
{\displaystyle {}M={\begin{pmatrix}0&0&0&\ldots &1\\1&0&0&\ldots &0\\0&1&0&\ldots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&0&\ldots &1&0\end{pmatrix}}\,.}
Determine the
characteristic polynomial
χ
M
{\displaystyle {}\chi _{M}}
M
{\displaystyle {}M}
Show that
P
=
X
−
1
{\displaystyle {}P=X-1}
χ
M
{\displaystyle {}\chi _{M}}
χ
M
=
P
Q
.
{\displaystyle {}\chi _{M}=PQ\,.}
Determine
P
(
M
)
{\displaystyle {}P(M)}
Q
(
M
)
{\displaystyle {}Q(M)}
Determine
kern
P
(
M
)
{\displaystyle {}\operatorname {kern} P(M)}
kern
Q
(
M
)
{\displaystyle {}\operatorname {kern} Q(M)}
Show that for a
diagonalizable mapping
φ
:
V
⟶
V
{\displaystyle \varphi \colon V\longrightarrow V}
and every
λ
∈
K
{\displaystyle {}\lambda \in K}
Eig
λ
(
φ
)
=
GeEig
λ
(
φ
)
{\displaystyle {}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {GeEig} _{\lambda }(\varphi )\,}
holds.
Let
φ
:
V
⟶
V
{\displaystyle \varphi \colon V\longrightarrow V}
be a
trigonalizable
mapping. Show that
φ
{\displaystyle {}\varphi }
diagonalizable
if and only if for every
λ
∈
K
{\displaystyle {}\lambda \in K}
Eig
λ
(
φ
)
=
GeEig
λ
(
φ
)
{\displaystyle {}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {GeEig} _{\lambda }(\varphi )\,}
holds.
Let
φ
:
V
⟶
V
{\displaystyle \varphi \colon V\longrightarrow V}
be a
trigonalizable
endomorphism ,
and let
V
=
H
1
⊕
⋯
⊕
H
m
{\displaystyle {}V=H_{1}\oplus \cdots \oplus H_{m}\,}
denote the
direct sum decomposition
into
generalized eigenspaces
in the sense of
Lemma 26.14
.
Show that there exists a
φ
{\displaystyle {}\varphi }
invariant flag
V
i
{\displaystyle {}V_{i}}
H
1
,
H
1
⊕
H
2
,
…
,
H
1
⊕
⋯
⊕
H
j
{\displaystyle H_{1},H_{1}\oplus H_{2},\ldots ,H_{1}\oplus \cdots \oplus H_{j}}
for
j
=
1
,
…
,
m
{\displaystyle {}j=1,\ldots ,m}
Hand-in-exercises Determine in
C
[
X
]
{\displaystyle {}\mathbb {C} [X]}
greatest common divisor
of the polynomials
X
3
+
(
2
−
i
)
X
2
+
4
{\displaystyle {}X^{3}+(2-{\mathrm {i} })X^{2}+4}
(
3
−
i
)
X
2
+
5
X
−
3
{\displaystyle {}(3-{\mathrm {i} })X^{2}+5X-3}
Determine the greatest common divisor of
4199
,
2431
{\displaystyle {}4199,2431}
3553
{\displaystyle {}3553}
We consider a digital clock, which exhibits
24
{\displaystyle {}24}
60
{\displaystyle {}60}
60
{\displaystyle {}60}
11
{\displaystyle {}11}
11
{\displaystyle {}11}
11
{\displaystyle {}11}
Let
K
{\displaystyle {}K}
field ,
let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space ,
and let
φ
:
V
⟶
V
{\displaystyle \varphi \colon V\longrightarrow V}
a
linear mapping . Let
P
∈
K
[
X
]
{\displaystyle {}P\in K[X]}
kern
(
P
(
φ
)
)
{\displaystyle \operatorname {kern} {\left(P(\varphi )\right)}}
is an
φ
{\displaystyle {}\varphi }
invariant
linear subspace.
Determine the
generalized eigenspaces
of the matrix
(
4
0
0
5
0
0
0
1
3
0
6
0
0
0
1
0
0
0
0
0
0
5
2
7
0
0
0
0
5
1
0
0
0
0
0
5
)
.
{\displaystyle {\begin{pmatrix}4&0&0&5&0&0\\0&1&3&0&6&0\\0&0&1&0&0&0\\0&0&0&5&2&7\\0&0&0&0&5&1\\0&0&0&0&0&5\end{pmatrix}}.}
![{\displaystyle {}\mathbb {Q} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05854fb7f5090d865105d7297fa178c8eb802fbf)
![{\displaystyle {}\mathbb {R} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52a170b822ae65cbe9d1cb122f493fcd88305b7f)
![{\displaystyle {}{\mathbb {F} }_{5}[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/331da43c9199a772b8d62e29d2918754f18351de)
![{\displaystyle {}K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7861df9df3d7463835d927c4d0d4c27f5e2f8416)
![{\displaystyle {}F,G\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2554b15433a4f1cc8f6ec612b0369190dbb2bf)
![{\displaystyle {}L[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d5815a5136aa33922f4b4abca07494764a41de7)
![{\displaystyle {}\mathbb {C} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5a91ede960f75df1b6b90029496a19f3b38d14)
![{\displaystyle {}P\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4692e3ffcbd66e900f8975113ff3f32f1cabc21a)
