Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 26/refcontrol

Exercise for the break

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Determine the greatest common divisorMDLD/greatest common divisor (polynomial) of ${}X^{2}-1$ and ${}X^{3}-1$, and a representation of it.

Exercises

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Determine the greatest common divisorMDLD/greatest common divisor (polynomial) of ${}X-1$ and ${}X-2$, and a representation of it.

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Determine the greatest common divisorMDLD/greatest common divisor (polynomial) of ${}X^{4}-X^{3}+7X^{2}-5X+11$ and ${}X^{3}-6X^{2}+4X+6$, and a representation of it.

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Determine in ${}\mathbb {Q} [X]$, using the Euclidean algorithm, the greatest common divisorMDLD/greatest common divisor (polynomial) of the polynomials ${}P=X^{3}+2X^{2}+5X+2$ and ${}Q=X^{2}+4X-3$.

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Determine in ${}\mathbb {Q} [X]$, using the Euclidean algorithm, the greatest common divisorMDLD/greatest common divisor (polynomial) of the polynomials ${}P=X^{9}-1$ and ${}Q=X^{3}-1$.

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Determine in ${}\mathbb {R} [X]$, using the Euclidean algorithm, the greatest common divisorMDLD/greatest common divisor (polynomial) of the polynomials ${}P=X^{3}+\pi X^{2}+7$ and ${}Q=X^{2}+{\sqrt {7}}X-{\sqrt {2}}$.

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Determine in ${}{\mathbb {F} }_{5}[X]$, using the Euclidean algorithm, the greatest common divisorMDLD/greatest common divisor (polynomial) of the polynomials ${}P=X^{4}+3X^{3}+X^{2}+4X+2$ and ${}Q=2X^{3}+4X^{2}+X+3$.

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Let ${}n\geq 2$.

1. Perform in ${}\mathbb {Q} [X]$ the division with remainderMDLD/division with remainder (polynomial) ${}P$ divided by ${}Q$ for the polynomials ${}P=X^{n-1}+X^{n-2}+\cdots +X^{2}+X+1$ and ${}T=X-1$.
2. Find a representation of ${}1$ as a linear combination of these polynomials.

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Let ${}K$ be a field,MDLD/field and let ${}K[X]$ be the polynomial ringMDLD/polynomial ring over ${}K$. Let ${}F,G\in K[X]$ be polynomials. Let ${}K\subseteq L$ be a field extension.MDLD/field extension Show that ${}F$ is a divisor of ${}G$ in ${}K[X]$ if and only if ${}F$ is a divisor of ${}G$ in ${}L[X]$.

The following exercises refer to the Euclidean algorithm for the integers, which works completely analogous to the Euclidean algorithm for polynomials. We first show why the Lemma of Bezout also holds for integers. ===Exercise Exercise 26.10

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Prove the Lemma of Bezout for integers ${}a_{1},\ldots ,a_{n}$.

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The delivery company "bucket without bugs“ is specialized in the transport of water. It has three buckets, one with ${}77$ liter, one with ${}91$ liter, and one with ${}143$ liter. The buckets do not have any marking. The company gets the request to transport, altogether, exactly one liter of water from the North Sea to the Baltic Sea. How can the company achieve this request?

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Determine in ${}\mathbb {Z}$, using the Euclidean algorithm, the greatest common divisorMDLD/greatest common divisor (Z) of ${}1071$ and ${}1029$.

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Determine in ${}\mathbb {Z}$, using the Euclidean algorithm, the greatest common divisorMDLD/greatest common divisor (Z) of ${}2956$ and ${}2444$.

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Determine in ${}\mathbb {Z}$, using the Euclidean algorithm, the greatest common divisorMDLD/greatest common divisor (Z) of ${}3146$ and ${}1515$. Find also a representation of the greatest common divisor as a linear combination with these numbers.

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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 ${}1$ with just one couple, whose age is one month. Let ${}f_{n}$ denote the number of couples of rabbits in the ${}n$-th month, hence ${}f_{1}=1$, ${}f_{2}=1$. Prove, by induction, the recursive formula

${}f_{n+2}=f_{n+1}+f_{n}\,.$

This sequence is called the sequence of the Fibonacci numbers. How many of the ${}f_{n}$ couples are in the ${}n$-th month able to reproduce?

The Fibonacci-numbers start with ${}1,1,2,3,5,8,13,21,34,\ldots$.

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Apply to two consecutive Fibonacci numbersMDLD/Fibonacci numbers the Euclidean algorithm.MDLD/Euclidean algorithm (Z) What can you observe?

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Determine the kernelsMDLD/kernels (linear) of the powers ${}M^{i}$ of the matrix

${}M={\begin{pmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{pmatrix}}\,.$

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Determine the kernels of the powers ${}M^{i}$ of the matrix

${}M={\begin{pmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&0\\0&0&0&0&1\\0&0&0&0&0\end{pmatrix}}\,.$

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Let

${}M={\begin{pmatrix}3&1&7\\0&3&5\\0&0&3\end{pmatrix}}\,.$

Determine the kernelsMDLD/kernels (linear) of the powers

${\left(3E_{3}-M\right)}^{i}.$

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Determine the generalized eigenspacesMDLD/generalized eigenspaces of the matrix

${\begin{pmatrix}1&1&0&0\\0&1&0&0\\0&0&2&3\\0&0&0&2\end{pmatrix}}.$

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Let ${}\pi$ be a cycleMDLD/cycle (permutation) of length ${}n$, and let ${}M$ denote the corresponding permutation matrix,MDLD/permutation matrix that is,

${}M={\begin{pmatrix}0&0&0&\ldots &1\\1&0&0&\ldots &0\\0&1&0&\ldots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&0&\ldots &1&0\end{pmatrix}}\,.$

1. Determine the characteristic polynomialMDLD/characteristic polynomial ${}\chi _{M}$ of ${}M$.
2. Show that ${}P=X-1$ divides ${}\chi _{M}$, and compute the factorization

   ${}\chi _{M}=PQ\,.$

3. Determine ${}P(M)$ and ${}Q(M)$.
4. Determine ${}\operatorname {kern} P(M)$ and ${}\operatorname {kern} Q(M)$.

===Exercise Exercise 26.22

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Show that for a diagonalizable mappingMDLD/diagonalizable mapping

$\varphi \colon V\longrightarrow V$

and every ${}\lambda \in K$, the equality

${}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {GeEig} _{\lambda }(\varphi )\,$

holds.

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Let

$\varphi \colon V\longrightarrow V$

be a trigonalizableMDLD/trigonalizable mapping. Show that ${}\varphi$ is diagonalizableMDLD/diagonalizable if and only if for every ${}\lambda \in K$, the equality

${}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {GeEig} _{\lambda }(\varphi )\,$

holds.

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Let

$\varphi \colon V\longrightarrow V$

be a trigonalizableMDLD/trigonalizable endomorphism,MDLD/endomorphism and let

${}V=H_{1}\oplus \cdots \oplus H_{m}\,$

denote the direct sum decompositionMDLD/direct sum decomposition into generalized eigenspacesMDLD/generalized eigenspaces in the sense of Lemma 26.14 . Show that there exists a ${}\varphi$-invariant flagMDLD/invariant flag ${}V_{i}$ such that, in this flag, the linear subspaces

$H_{1},H_{1}\oplus H_{2},\ldots ,H_{1}\oplus \cdots \oplus H_{j}$

for ${}j=1,\ldots ,m$ occur.

Hand-in-exercises

Exercise (3 marks) Create referencenumber

Determine in ${}\mathbb {C} [X]$, using the Euclidean algorithm, the greatest common divisorMDLD/greatest common divisor (polynomial) of the polynomials ${}X^{3}+(2-{\mathrm {i} })X^{2}+4$ and ${}(3-{\mathrm {i} })X^{2}+5X-3$.

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Determine the greatest common divisor of ${}4199,2431$ and ${}3553$, and a representation of it as a linear combination of the given numbers.

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We consider a digital clock, which exhibits ${}24$ hours, ${}60$ minutes and ${}60$ seconds. However, during carnival, it does not run in steps of seconds, but it adds, starting from the zero position, in every step ${}11$ hours, ${}11$ minutes and ${}11$ seconds. Does, with this way of counting, every possible digital display appear? After how many steps does the zero position reoccur for the first time?

===Exercise (3 marks) Exercise 26.28

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Let ${}K$ be a field,MDLD/field let ${}V$ be a ${}K$-vector space,MDLD/vector space and let

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping Let ${}P\in K[X]$ be a polynomial. Show that

$\operatorname {kern} {\left(P(\varphi )\right)}$

is an ${}\varphi$-invariantMDLD/invariant (endomorphism) linear subspace.

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Determine the generalized eigenspacesMDLD/generalized eigenspaces of the matrix

${\begin{pmatrix}4&0&0&5&0&0\\0&1&3&0&6&0\\0&0&1&0&0&0\\0&0&0&5&2&7\\0&0&0&0&5&1\\0&0&0&0&0&5\end{pmatrix}}.$

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