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

Exercise for the break

Exercise

Determine the greatest common divisor of $X^2-1$ and $X^3-1$, and a representation of it.

Exercises

Exercise

Determine the greatest common divisor of $X-1$ and $X-2$, and a representation of it.

Exercise

Determine the greatest common divisor of $X^4-X^3+7X^2-5X+11$ and $X^3-6X^2+4X+6$, and a representation of it.

Exercise

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

Exercise

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

Exercise

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

Exercise

Determine in ${𝔽}_5[X]$, using the Euclidean algorithm, the greatest common divisor of the polynomials $P=X^4+3X^3+X^2+4X+2$ and $Q=2X^3+4X^2+X+3$.

Exercise

Let $n\ge 2$.

1. Perform in $\mathbb{Q}[X]$ the division with remainder $P$ 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.

Exercise

Let $K$ be a field, and let $K[X]$ be the polynomial ring over $K$. Let $F,G\in K[X]$ be polynomials. Let $K\subseteq L$ be a 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

Prove the Lemma of Bezout for integers $a_1,\dots ,a_n$.

Exercise

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?

Exercise

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

Exercise

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

Exercise

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

Exercise

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 $1,1,2,3,5,8,13,21,34,\dots$.

Exercise

Apply to two consecutive Fibonacci numbers the Euclidean algorithm. What can you observe?

Exercise

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

$M=\left(\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right).$

Exercise

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

$M=\left(\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right).$

Exercise

Let

$M=\left(\begin{array}{ccc}3& 1& 7\\ 0& 3& 5\\ 0& 0& 3\end{array}\right).$

Determine the kernels of the powers

${(3E_3-M)}^i.$

Exercise

Determine the generalized eigenspaces of the matrix

$\left(\begin{array}{cccc}1& 1& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 2& 3\\ 0& 0& 0& 2\end{array}\right).$

Exercise

Let $\pi$ be a cycle of length $n$, and let $M$ denote the corresponding permutation matrix, that is,

$M=\left(\begin{array}{ccccc}0& 0& 0& \dots & 1\\ 1& 0& 0& \dots & 0\\ 0& 1& 0& \dots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& 0& \dots & 1& 0\end{array}\right).$

1. Determine the 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 $\mathrm{kern}P(M)$ and $\mathrm{kern}Q(M)$.

Exercise

Let $\phi :V\to V$ be a linear mapping on a finite-dimensional $K$-vector space $V$, and let

${\chi }_{\phi }=P\cdot Q$

be a factorization of the characteristic polynomial into coprime polynomials $P,Q\in K[X]$ with the corresponding direct sum decomposition

$V=\mathrm{kern}P(\phi )\oplus \mathrm{kern}Q(\phi )$

into $\phi$-invariant linear subspaces. Let $U\subseteq V$ be a $\phi$-invariant linear subspace. Show that

$U=(U\cap \mathrm{kern}P(\phi ))\oplus (U\cap \mathrm{kern}Q(\phi ))$

holds.

Exercise

Show that for a diagonalizable mapping

$\phi :V⟶V$

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

${\mathrm{Eig}}_{\lambda }\left(\phi \right)={\mathrm{GeEig}}_{\lambda }(\phi )$

holds.

Exercise

Let

$\phi :V⟶V$

be a trigonalizable mapping. Show that $\phi$ is diagonalizable if and only if for every $\lambda \in K$, the equality

${\mathrm{Eig}}_{\lambda }\left(\phi \right)={\mathrm{GeEig}}_{\lambda }(\phi )$

holds.

Exercise

Let

$\phi :V⟶V$

be a trigonalizable endomorphism, and let

$V=H_1\oplus \cdots \oplus H_m$

denote the direct sum decomposition into generalized eigenspaces in the sense of Lemma 26.14 . Show that there exists a $\phi$-invariant flag $V_i$ such that, in this flag, the linear subspaces

$H_1,H_1\oplus H_2,\dots ,H_1\oplus \cdots \oplus H_j$

for $j=1,\dots ,m$ occur.

Hand-in-exercises

Exercise (3 marks)

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

Exercise (4 marks)

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

Exercise (5 marks)

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)

Let $K$ be a field, let $V$ be a $K$-vector space, and let

$\phi :V⟶V$

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

$\mathrm{kern}(P(\phi ))$

is an $\phi$-invariant linear subspace.

Exercise (4 marks)

Determine the generalized eigenspaces of the matrix

$\left(\begin{array}{cccccc}4& 0& 0& 5& 0& 0\\ 0& 1& 3& 0& 6& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 5& 2& 7\\ 0& 0& 0& 0& 5& 1\\ 0& 0& 0& 0& 0& 5\end{array}\right).$

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