## Find a Polynomial Satisfying the Given Conditions on Derivatives

## Problem 87

Find a cubic polynomial

\[p(x)=a+bx+cx^2+dx^3\]
such that $p(1)=1, p'(1)=5, p(-1)=3$, and $ p'(-1)=1$.

Find a cubic polynomial

\[p(x)=a+bx+cx^2+dx^3\]
such that $p(1)=1, p'(1)=5, p(-1)=3$, and $ p'(-1)=1$.

Consider a polynomial

\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.

Define the matrix

\[A=\begin{bmatrix}

0 & 0 & \dots & 0 &-a_0 \\

1 & 0 & \dots & 0 & -a_1 \\

0 & 1 & \dots & 0 & -a_2 \\

\vdots & & \ddots & & \vdots \\

0 & 0 & \dots & 1 & -a_{n-1}

\end{bmatrix}.\]

Then prove that the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$.

The matrix is called the ** companion matrix** of the polynomial $p(x)$.

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Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix.

Show that the matrix $A$ is diagonalizable.

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Let $f(x)$ be an irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be any polynomial in $F[x]$.

Show that the degree of each irreducible factor of the composite polynomial $f(g(x))$ is divisible by $n$.

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Show that the polynomial $x^3-\sqrt{2}$ is irreducible over the field $\Q(\sqrt{2})$.

Add to solve laterLet $G$ be a group of order $|G|=pqr$, where $p,q,r$ are prime numbers such that $p<q<r$.

Show that $G$ has a normal subgroup of order either $p,q$ or $r$.

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Let $V$ be a finite dimensional vector space over a field $K$ and let $\End (V)$ be the vector space of linear transformations from $V$ to $V$.

Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$ be a basis for $V$.

Show that the map $\phi:\End (V) \to V^{\oplus n}$ defined by $f\mapsto (f(\mathbf{v}_1), \dots, f(\mathbf{v}_n))$ is an isomorphism.

Here $V^{\oplus n}=V\oplus \dots \oplus V$, the direct sum of $n$ copies of $V$.

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Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.

That is,

\begin{equation*}

V:=\left\{ A=\begin{bmatrix}

a_{11} & 0 & \dots & 0 \\

0 &a_{22} & \dots & 0 \\

0 & 0 & \ddots & \vdots \\

0 & 0 & \dots & a_{nn}

\end{bmatrix} \quad \middle| \quad

\begin{array}{l}

a_{11}, \dots, a_{nn} \in \C,\\

\tr(A)=0 \\

\end{array}

\right\}

\end{equation*}

Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.

**(a)** Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)

**(b)** Show that matrices

\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.

**(c)** Find the dimension of $V$.

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Determine whether the following sentence is True or False.

(*Purdue University Linear Algebra Exam*)

A square matrix $A$ is called ** nilpotent** if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.

**(a)** If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent.

**(b)** Let $P$ be an invertible $n \times n$ matrix and let $N$ be a nilpotent $n\times n$ matrix. Is the product $PN$ nilpotent? If so, prove it. If not, give a counterexample.

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Let A be the matrix

\[\begin{bmatrix}

1 & -1 & 0 \\

0 &1 &-1 \\

0 & 0 & 1

\end{bmatrix}.\]
Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.

(*The Ohio State University Linear Algebra Exam*)

Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$.

**(a)** Show that $B=\{1, \sqrt{2}\}$ is a basis for the vector space $V$ over $\Q$.

**(b)** Let $\alpha=a+b\sqrt{2} \in V$, and let $T_{\alpha}: V \to V$ be the map defined by

\[ T_{\alpha}(x+y\sqrt{2}):=(ax+2by)+(ay+bx)\sqrt{2}\in V\]
for any $x+y\sqrt{2} \in V$.

Show that $T_{\alpha}$ is a linear transformation.

**(c)** Let $\begin{bmatrix}

x \\

y

\end{bmatrix}_B=x+y \sqrt{2}$.

Find the matrix $T_B$ such that

\[ T_{\alpha} (x+y \sqrt{2})=\left( T_B\begin{bmatrix}

x \\

y

\end{bmatrix}\right)_B,\]
and compute $\det T_B$.

(*The Ohio State University, Linear Algebra Exam*)

**(a)** Show that if a group $G$ has the following order, then it is not simple.

- $28$
- $496$
- $8128$

**(b) **Show that if the order of a group $G$ is equal to an even * perfect number* then the group is not simple.

Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers.

Show that exponential functions

\[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\]
are linearly independent over $\R$.

**(a)** Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that the entries of the matrix $A$ satisfy the following relation.

\[|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|\]
for all $1 \leq i \leq n$.

Show that the matrix $A$ is nonsingular.

**(b) **Let $B=(b_{ij})$ be an $n \times n$ matrix whose entries satisfy the relation

\[ |b_{i\,i}|=1 \hspace{0.5cm} \text{ and }\hspace{0.5cm} |b_{ij}|<\frac{1}{n-1}\]
for all $i$ and $j$ with $i \neq j$.

Prove that the matrix $B$ is nonsingular.

**(c)**

Determine whether the following matrix is nonsingular or not.

\[C=\begin{bmatrix}

\pi & e & e^2/2\pi^2 \\[5 pt]
e^2/2\pi^2 &\pi &e \\[5pt]
e & e^2/2\pi^2 & \pi

\end{bmatrix},\]
where $\pi=3.14159\dots$, and $e=2.71828\dots$ is Euler’s number (or Napier’s constant).

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Let $P_2(\R)$ be the vector space over $\R$ consisting of all polynomials with real coefficients of degree $2$ or less.

Let $B=\{1,x,x^2\}$ be a basis of the vector space $P_2(\R)$.

For each linear transformation $T:P_2(\R) \to P_2(\R)$ defined below, find the matrix representation of $T$ with respect to the basis $B$. For $f(x)\in P_2(\R)$, define $T$ as follows.

**(a)** \[T(f(x))=\frac{\mathrm{d}^2}{\mathrm{d}x^2} f(x)-3\frac{\mathrm{d}}{\mathrm{d}x}f(x)\]

**(b)** \[T(f(x))=\int_{-1}^1\! (t-x)^2f(t) \,\mathrm{d}t\]

**(c)** \[T(f(x))=e^x \frac{\mathrm{d}}{\mathrm{d}x}(e^{-x}f(x))\]

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Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$.

**(a)** If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not.

**(b)** Is $3\mathbf{v}$ an eigenvector of $A$? If so, what is the corresponding eigenvalue? If not, explain why not.

(*Stanford University, Linear Algebra Exam*)

Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation

\[HF-FH=-2F.\]

**(a)** Find the trace of the matrix $F$.

**(b)** Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ such that $F^N\mathbf{v}=\mathbf{0}$.

Let $H$ and $E$ be $n \times n$ matrices satisfying the relation

\[HE-EH=2E.\]
Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$.

Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then prove that

\[E\mathbf{x}=\mathbf{0}.\]