Let $R$ be a commutative ring with $1$. Prove that the principal ideal $(x)$ generated by the element $x$ in the polynomial ring $R[x]$ is a prime ideal if and only if $R$ is an integral domain.
Prove also that the ideal $(x)$ is a maximal ideal if and only if $R$ is a field.
Add to solve later Let $R$ be a ring with unit $1\neq 0$.
Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$.
(Do not assume that the ring $R$ is commutative.)
Add to solve later Let $G$ be a group. Assume that $H$ and $K$ are both normal subgroups of $G$ and $H \cap K=1$. Then for any elements $h \in H$ and $k\in K$, show that $hk=kh$.
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Add to solve later Let $G$ be a group and let $A$ be an abelian subgroup of $G$ with $A \triangleleft G$.
(That is, $A$ is a normal subgroup of $G$.)
If $B$ is any subgroup of $G$, then show that
\[A \cap B \triangleleft AB.\]
Add to solve later Find the value(s) of $h$ for which the following set of vectors
\[\left \{ \mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
h \\
1 \\
-h
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
1 \\
2h \\
3h+1
\end{bmatrix}\right\}\]
is linearly independent.
(Boston College, Linear Algebra Midterm Exam Sample Problem)
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Add to solve later Let $A$ be a $3 \times 3$ matrix.
Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have
\[A\mathbf{x}=\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}, A\mathbf{y}=\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}, A\mathbf{z}=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}.\]
Then find the value of the determinant of the matrix $A$.
Add to solve later Let
\[A=\begin{bmatrix}
1 & -1\\
2& 3
\end{bmatrix}.\]
Find the eigenvalues and the eigenvectors of the matrix
\[B=A^4-3A^3+3A^2-2A+8E.\]
(Nagoya University Linear Algebra Exam Problem)
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Add to solve later Prove that the matrix
\[A=\begin{bmatrix}
1 & 1.00001 & 1 \\
1.00001 &1 &1.00001 \\
1 & 1.00001 & 1
\end{bmatrix}\]
has one positive eigenvalue and one negative eigenvalue.
(University of California, Berkeley Qualifying Exam Problem)
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Add to solve later Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces
\[E_2=\Span\left \{\quad \begin{bmatrix}
1 \\
1 \\
1 \\
1
\end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix}
1 \\
2 \\
1 \\
1
\end{bmatrix},\quad \begin{bmatrix}
1 \\
1 \\
1 \\
2
\end{bmatrix} \quad\right\}.\]
Calculate $C^4 \mathbf{u}$ for $\mathbf{u}=\begin{bmatrix}
6 \\
8 \\
6 \\
9
\end{bmatrix}$ if possible. Explain why if it is not possible!
(The Ohio State University Linear Algebra Exam Problem)
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Add to solve later Denote by $i$ the square root of $-1$.
Let
\[R=\Z[i]=\{a+ib \mid a, b \in \Z \}\]
be the ring of Gaussian integers.
We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to
\[N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.\]
Here $\bar{\alpha}$ is the complex conjugate of $\alpha$.
Then show that an element $\alpha \in R$ is a unit if and only if the norm $N(\alpha)=\pm 1$.
Also, determine all the units of the ring $R=\Z[i]$ of Gaussian integers.
Add to solve later Let $A$ be an $n\times n$ matrix. Suppose that $\lambda_1, \lambda_2$ are distinct eigenvalues of the matrix $A$ and let $\mathbf{v}_1, \mathbf{v}_2$ be eigenvectors corresponding to $\lambda_1, \lambda_2$, respectively.
Show that the vectors $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent.
Add to solve later Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$.
Is it true that
\[\det(A+B)=\det(A)+\det(B)?\]
If so, then give a proof. If not, then give a counterexample.
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Add to solve later Let $A=(a_{ij})$ be an $n \times n$ matrix.
We say that $A=(a_{ij})$ is a right stochastic matrix if each entry $a_{ij}$ is nonnegative and the sum of the entries of each row is $1$. That is, we have
\[a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1\]
for $1 \leq i, j \leq n$.
Let $A=(a_{ij})$ be an $n\times n$ right stochastic matrix. Then show the following statements.
(a)The stochastic matrix $A$ has an eigenvalue $1$.
(b) The absolute value of any eigenvalue of the stochastic matrix $A$ is less than or equal to $1$.
Add to solve later Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix.
If
\[P^{-1}AP=\begin{bmatrix}
1 & 2 & 3 \\
0 &4 &5 \\
0 & 0 & 6
\end{bmatrix},\]
then find all the eigenvalues of the matrix $A^2$.
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Add to solve later Let $A$ be an $n \times n$ matrix. Suppose that the matrix $A^2$ has a real eigenvalue $\lambda>0$. Then show that either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of the matrix $A$.
Add to solve later Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$.
Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$.
Show that the vectors $\mathbf{x}, T\mathbf{x}, T^2\mathbf{x}$ form a basis for $\R^3$.
(The Ohio State University Linear Algebra Exam Problem)
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Add to solve later Suppose that $\begin{bmatrix}
1 \\
1
\end{bmatrix}$ is an eigenvector of a matrix $A$ corresponding to the eigenvalue $3$ and that $\begin{bmatrix}
2 \\
1
\end{bmatrix}$ is an eigenvector of $A$ corresponding to the eigenvalue $-2$.
Compute $A^2\begin{bmatrix}
4 \\
3
\end{bmatrix}$.
(Stanford University Linear Algebra Exam Problem)
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Add to solve later Suppose the following information is known about a $3\times 3$ matrix $A$.
\[A\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix}=6\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix},
\quad
A\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}=3\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}, \quad
A\begin{bmatrix}
2 \\
-1 \\
0
\end{bmatrix}=3\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}.\]
(a) Find the eigenvalues of $A$.
(b) Find the corresponding eigenspaces.
(c) In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.
Is $A$ a diagonalizable matrix?
Is $A$ an invertible matrix?
Is $A$ an idempotent matrix?
(Johns Hopkins University Linear Algebra Exam)
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Add to solve later Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.
Add to solve later 
