## $\sqrt[m]{2}$ is an Irrational Number

## Problem 179

Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.

Add to solve laterProve that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.

Add to solve laterLet

\[\begin{bmatrix}

0 & 0 & 1 \\

1 &0 &0 \\

0 & 1 & 0

\end{bmatrix}.\]

**(a)** Find the characteristic polynomial and all the eigenvalues (real and complex) of $A$. Is $A$ diagonalizable over the complex numbers?

**(b)** Calculate $A^{2009}$.

(*Princeton University, Linear Algebra Exam*)

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Let $R$ be a principal ideal domain (PID). Let $a\in R$ be a non-unit irreducible element.

Then show that the ideal $(a)$ generated by the element $a$ is a maximal ideal.

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Let $A$ be an $n \times n$ matrix. We say that $A$ is **idempotent** if $A^2=A$.

**(a)** Find a nonzero, nonidentity idempotent matrix.

**(b)** Show that eigenvalues of an idempotent matrix $A$ is either $0$ or $1$.

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

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Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$.

Show that $P$ is a maximal ideal in $R$.

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Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent:

**(a)** The ideal $P$ is a prime ideal.

**(b)** For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.

Let $R$ be a commutative ring. An ideal $I$ of $R$ is said to be **irreducible** if it cannot be written as an intersection of two ideals of $R$ which are strictly larger than $I$.

Prove that if $\frakp$ is a prime ideal of the commutative ring $R$, then $\frakp$ is irreducible.

Add to solve laterLet $R$ be a commutative ring.

Then prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal of $R$.

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Let $R$ be a commutative ring with $1 \neq 0$.

An element $a\in R$ is called **nilpotent** if $a^n=0$ for some positive integer $n$.

Then prove that if $a$ is a nilpotent element of $R$, then $1-ab$ is a unit for all $b \in R$.

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Prove that the rings $2\Z$ and $3\Z$ are not isomorphic.

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Find all the values of $x$ so that the following matrix $A$ is a singular matrix.

\[A=\begin{bmatrix}

x & x^2 & 1 \\

2 &3 &1 \\

0 & -1 & 1

\end{bmatrix}.\]

Let

\[A=\begin{bmatrix}

1 & -x & 0 & 0 \\

0 &1 & -x & 0 \\

0 & 0 & 1 & -x \\

0 & 1 & 0 & -1

\end{bmatrix}\]
be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.

Let $G, G’$ be groups. Suppose that we have a surjective group homomorphism $f:G\to G’$.

Show that if $G$ is an abelian group, then so is $G’$.

Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.)

**(a)** Prove that the subset $W$ is a subspace of $V$.

**(b)** Find the dimension of $W$.

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

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Let $P_2$ be the vector space of all polynomials of degree two or less.

Consider the subset in $P_2$

\[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\]
where

\begin{align*}

&p_1(x)=1, &p_2(x)=x^2+x+1, \\

&p_3(x)=2x^2, &p_4(x)=x^2-x+1.

\end{align*}

**(a)** Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$.

**(b)** Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$.

**(c)** For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors.

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

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Let $T:\R^4 \to \R^3$ be a linear transformation defined by

\[ T\left (\, \begin{bmatrix}

x_1 \\

x_2 \\

x_3 \\

x_4

\end{bmatrix} \,\right) = \begin{bmatrix}

x_1+2x_2+3x_3-x_4 \\

3x_1+5x_2+8x_3-2x_4 \\

x_1+x_2+2x_3

\end{bmatrix}.\]

**(a)** Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.

**(b)** Find a basis for the null space of $T$.

**(c)** Find the rank of the linear transformation $T$.

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

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Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism.

Then show that there exists an integer $a$ such that

\[f(n)=an\]
for any integer $n$.

Let $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ are vectors in $\R^n$. Suppose that vectors $\mathbf{u}_1$, $\mathbf{u}_2$ are orthogonal and the norm of $\mathbf{u}_2$ is $4$ and $\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number $a$ in $\mathbf{u_1}=\mathbf{u_2}+a\mathbf{u}_3$.

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

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Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$.

Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.

Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$.

Then show that $H$ is a subgroup of $G$.

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