# 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 $\mathbb {R} [X]$, 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 ${\mathbb {F} }_{5}[X]$, 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

Polynomial ring/X^(n-1)+...+1/X-1/Representation of 1/Exercise

### Exercise

Polynomial ring/Field/One variable/Field extension/Divisor relation/Exercise
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

Divisibility theory (Z)/Lemma of Bezout/Fact/Proof/Exercise

### Exercise

Generated subgroup/77,91,143/Representation of 1/Bucket/Exercise

### Exercise

Euclidean algorithm (Z)/Gcd/1071 and 1029/Exercise

### Exercise

Euclidean algorithm (Z)/Gcd/2956 and 2444/Exercise

### Exercise

Euclidean algorithm (Z)/Gcd/3146 and 1515/Representation/Exercise

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

### Exercise

Fibonacci-numbers/Euclidean algorithm/Exercise

### Exercise

Matrix/Nilpotent/Jordan/Successive kernels/1/Exercise

### Exercise

Matrix/Nilpotent/Jordan/Successive kernels/2/Exercise

### Exercise

Matrix/317035003/Successive kernels to 3Id-M/Exercise

### Exercise

Matrix/1100010000230002/Generalized eigenspaces/Exercise

### Exercise

Permutation matrix/Cycle/Characteristic polynomial/Factorizaton/Sum decomposition/Exercise

### Exercise

Diagonalizable/Generalized eigenspace/Exercise

### Exercise

Trigonalizable/Diagonalizable/Eigen- and generalized eigenspace/Exercise

### Exercise

Trigonalizable/Flag via generalized eigenspaces/Exercise
    Hand-in-exercises

### Exercise (3 marks)

Determine in $\mathbb {C} [X]$, 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 dispaly 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
    $\varphi \colon V\longrightarrow V$
a linear mapping. Let $P\in K[X]$ be a polynomial. Show that
    $\operatorname {kern} {\left(P(\varphi )\right)}$
is an $\varphi$-invariant linear subspace.

### Exercise (4 marks)

Determine the generalized eigenspaces of the matrix
    ${\begin{pmatrix}4&0&0&5&0&0\\0&1&3&0&6&0\\0&0&1&0&0&0\\0&0&0&5&2&7\\0&0&0&0&5&1\\0&0&0&0&0&5\end{pmatrix}}.$