## Is the Set of Nilpotent Element an Ideal?

## Problem 620

Is it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$?

If so, prove it. Otherwise give a counterexample.

Add to solve laterIs it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$?

If so, prove it. Otherwise give a counterexample.

Add to solve laterLet $R$ be a commutative ring with $1$ such that every element $x$ in $R$ is idempotent, that is, $x^2=x$. (Such a ring is called a **Boolean ring**.)

**(a)** Prove that $x^n=x$ for any positive integer $n$.

**(b)** Prove that $R$ does not have a nonzero nilpotent element.

Let $R$ be a commutative ring with $1$.

Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$.

Is it true that $A$ is also a Noetherian ring?

Suppose that $p$ is a prime number greater than $3$.

Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$.

**(a)** Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$.

**(b)** Determine the index $[G : S]$.

**(c)** Assume that $-1\notin S$. Then prove that for each $a\in G$ we have either $a\in S$ or $-a\in S$.

Let $R$ be a ring and assume that whenever $ab=ca$ for some elements $a, b, c\in R$, we have $b=c$.

Then prove that $R$ is a commutative ring.

Add to solve laterLet $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$.

Prove that the number of elements in $S$ is odd.

Add to solve laterLet $m$ and $n$ be positive integers such that $m \mid n$.

**(a)** Prove that the map $\phi:\Zmod{n} \to \Zmod{m}$ sending $a+n\Z$ to $a+m\Z$ for any $a\in \Z$ is well-defined.

**(b)** Prove that $\phi$ is a group homomorphism.

**(c)** Prove that $\phi$ is surjective.

**(d)** Determine the group structure of the kernel of $\phi$.

Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.

Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.

**(a)** Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$.

**(b)** Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$.

An $n\times n$ matrix $A$ is called **orthogonal** if $A^{\trans}A=I$.

Let $V$ be the vector space of all real $2\times 2$ matrices.

Consider the subset

\[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\]
Prove or disprove that $W$ is a subspace of $V$.

Let $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure below.

Find the matrix representation $A$ of the linear transformation $T$.

Add to solve later Let $A$ be a $2\times 2$ real symmetric matrix.

Prove that all the eigenvalues of $A$ are real numbers by considering the characteristic polynomial of $A$.

Let $A$ and $B$ be $n\times n$ matrices and assume that they commute: $AB=BA$.

Then prove that the matrices $A$ and $B$ share at least one common eigenvector.

Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less.

Let

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

\begin{align*}

p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\

p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3.

\end{align*}

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

**(b)** For each polynomial in $S$ that is not in $Q$, find the coordinate vector with respect to the basis $Q$.

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

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Let $V$ be a vector space and $B$ be a basis for $V$.

Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.

Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ with respect to the basis $B$.

After applying the elementary row operations to $A$, we obtain the following matrix in reduced row echelon form

\[\begin{bmatrix}

1 & 0 & 2 & 1 & 0 \\

0 & 1 & 3 & 0 & 1 \\

0 & 0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0 & 0

\end{bmatrix}.\]

**(a)** What is the dimension of $V$?

**(b)** What is the dimension of $\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$?

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

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Let $T:\R^2 \to \R^3$ be a linear transformation such that

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

3 \\

2

\end{bmatrix} \,\right)

=\begin{bmatrix}

1 \\

2 \\

3

\end{bmatrix} \text{ and }

T\left(\, \begin{bmatrix}

4\\

3

\end{bmatrix} \,\right)

=\begin{bmatrix}

0 \\

-5 \\

1

\end{bmatrix}.\]

**(a)** Find the matrix representation of $T$ (with respect to the standard basis for $\R^2$).

**(b)** Determine the rank and nullity of $T$.

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

Read solution

Let

\[A=\begin{bmatrix}

1 & -1 & 0 & 0 \\

0 &1 & 1 & 1 \\

1 & -1 & 0 & 0 \\

0 & 2 & 2 & 2\\

0 & 0 & 0 & 0

\end{bmatrix}.\]

**(a)** Find a basis for the null space $\calN(A)$.

**(b)** Find a basis of the range $\calR(A)$.

**(c)** Find a basis of the row space for $A$.

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

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Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$.

Consider the functions \[f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)\]
in $C[-2\pi, 2\pi]$.

Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly independent.

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

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Let $W$ be a subspace of $\R^4$ with a basis

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

1 \\

0 \\

1 \\

1

\end{bmatrix}, \begin{bmatrix}

0 \\

1 \\

1 \\

1

\end{bmatrix} \,\right\}.\]

Find an ortho**normal** basis of $W$.

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

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Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers.

Let

\[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix}

a & b\\

c& -a

\end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\]

**(a)** Show that $W$ is a subspace of $V$.

**(b)** Find a basis of $W$.

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

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

Read solution