Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 20

Exercise for the break

Exercise

Compute the result, when we replace in the polynomial

$X^{2}-5X+3$

the Variable ${}X$ by the ${}2\times 2$-matrix

${\begin{pmatrix}6&7\\-4&5\end{pmatrix}}.$

Exercises

Exercise *

Find a polynomial

${}f=a+bX+cX^{2}\,$

with ${}a,b,c\in \mathbb {R}$, such that the following conditions hold.

$f(-1)=2,\,f(1)=0,\,f(3)=5.$

Exercise

Find a polynomial

${}f=a+bX+cX^{2}+dX^{3}\,$

with ${}a,b,c,d\in \mathbb {R}$, such that the following conditions hold.

$f(0)=1,\,f(1)=2,\,f(2)=0,\,f(-1)=1.$

Exercise

Find, for each of the following three sets

$\{1,3,5,7,9,\ldots \},\,\{1,2,4,8,16,\ldots \},\,\{9,99,999,9999,99999,\ldots \}$

(of the form ${}\{a_{1},a_{2},a_{3},a_{4},a_{5},\ldots \}$) a polynomial

${}P(k)=c_{0}+c_{1}k+c_{2}k^{2}+c_{3}k^{3}+c_{4}k^{4}\,$

(with coefficients ${}c_{j}\in \mathbb {Q}$) such that

$P(1)=a_{1},\,P(2)=a_{2},\,P(3)=a_{3},\,P(4)=a_{4},\,P(5)=a_{5}$

holds.

Exercise

Let ${}K$ be a finite field with ${}q$ elements. Show that every function ${}\varphi \colon K\rightarrow K$ can be expressed in a unique way as a polynomial ${}P\in K[X]$ of degree ${}<q$.

Exercise

Let ${}K$ be a finite field with ${}q$ elements.

1. Show that the polynomial functions

   $\varphi _{d}\colon K\longrightarrow K,x\longmapsto x^{d},$

   with ${}0\leq d<q$ are linearly independent.

2. Show that the exponential functions

   $\psi _{b}\colon K\longrightarrow K,x\longmapsto b^{x},$

   with ${}0\leq b<q$ are linearly independent.

Exercise

Compute the result, when we replace in the polynomial

$2X^{3}-5X^{2}+7X-4$

the variable ${}X$ by the ${}2\times 2$-matrix

${\begin{pmatrix}4&1\\5&3\end{pmatrix}}.$

Exercise

Let

$f,g\colon V\longrightarrow V$

be endomorphisms on a ${}K$-vector space ${}V$, and let ${}P\in K[X]$ be a polynomial. Show that the equality

${}P(f\circ g)=P(f)\circ P(g)\,$

does not hold in general.

Exercise

For a ${}2\times 2$-matrix ${}M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$, let

${}P_{M}:=X^{2}-\operatorname {trace} {\left(M\right)}X+\det M\,.$

Show that ${}P_{M}(M)=0$ holds.

Exercise

Let ${}V$ be a finite-dimensional vector space over a field ${}K$, and let

$f\colon V\longrightarrow V$

denote a linear mapping. Show that the set

${\left\{P\in K[X]\mid P(f)=0\right\}}$

is a principal ideal in the polynomial ring ${}K[X]$, which is generated by the minimal polynomial ${}\mu _{f}$.

Exercise

Let ${}M$ be a matrix with minimal polynomial ${}X-a$. Show that ${}M$ is a homothety with factor ${}a$.

Exercise

We discuss the minimal polynomials of the elementary matrices.

a) Show that the minimal polynomial of an exchange matrix ${}V_{ij}$ is ${}X^{2}-1$.

b) Show that the minimal polynomial of a scaling elementary matrix ${}S_{k}(s)$ with ${}s\neq 1$ is

$X^{2}-(s+1)X+s.$

c) Show that the minimal polynomial of an addition matrix ${}A_{ij}(a)$ is of the form

$(X-1)^{k}.$

What is ${}k$?

Exercise

Let ${}V\neq 0$ be a finite-dimensional ${}K$-vector space, and let

$\varphi \colon V\longrightarrow V$

denote a projection. Show that the minimal polynomial of ${}\varphi$ is either ${}X$, or ${}X-1$, or ${}X(X-1)$.

Exercise

Let ${}K\subseteq L$ be a field extension. Let an ${}n\times n$-matrix ${}M$ over ${}K$ be given. Show that the minimal polynomial ${}P\in K[X]$ coincides with the minimal polynomial of ${}M$, considered as a matrix over ${}L$.

Exercise

Let ${}M$ be an ${}n\times n$-matrix over ${}K$ with the minimal polynomial ${}P\in K[X]$. Let

${}P=F_{1}\cdots F_{k}\,$

be a factorization into polynomials ${}F_{i}$ of positive degree. Show that ${}F_{i}(M)$ is not bijective.

Exercise

We consider the linear mapping

$\varphi \colon K^{(\mathbb {N} )}\longrightarrow K^{(\mathbb {N} )}$

given by ${}e_{n}\mapsto e_{n+1}$. Show that ${}\varphi$ is only annihilated by the zero polynomial.

Hand-in-exercises

Exercise (4 marks)

Find a polynomial ${}f$ of degree ${}\leq 3$ for which

$f(0)=-1,\,f(-1)=-3,\,f(1)=7,\,f(2)=21$

holds.

Exercise (3 marks)

Find a polynomial

${}f=a+bX+cX^{2}\,$

with ${}a,b,c\in \mathbb {C}$, such that the following conditions hold.

$f({\mathrm {i} })=1,\,f(1)=1+{\mathrm {i} },\,f(1-2{\mathrm {i} })=-{\mathrm {i} }.$

Exercise (3 marks)

Compute the result, when we replace in the polynomial

$-X^{3}+6X^{2}-6X+27,$

the variable ${}X$ by the ${}3\times 3$-matrix

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

Exercise (3 marks)

Let

$f\colon V\longrightarrow V$

be an endomorphism on a ${}K$-vector space ${}V$, and let

$g\colon V\longrightarrow V$

denote an isomorphism. Show that, for every polynomial ${}P\in K[X]$, the equality

${}gP(f)g^{-1}=P(gfg^{-1})\,$

holds.

<< | Linear algebra (Osnabrück 2024-2025)/Part I | >> PDF-version of this exercise sheet Lecture for this exercise sheet (PDF)