Matrix Operations with Transpose
Calculate the following expressions, using the following matrices:
\[A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}\]
(a) $A B^\trans + \mathbf{v} […]

Diagonalizable Matrix with Eigenvalue 1, -1
Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues.
Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix.
(Stanford University Linear Algebra Exam)
See below for a generalized problem.
Hint.
Diagonalize the […]

Galois Group of the Polynomial $x^2-2$
Let $\Q$ be the field of rational numbers.
(a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$?
(b) Find the Galois group of $f(x)$ over $\Q$.
Solution.
(a) The polynomial $f(x)=x^2-2$ is separable over $\Q$
The roots of the polynomial $f(x)$ are $\pm […]

The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD)
Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).
Proof.
Every element of the ring $\Z[\sqrt{5}]$ can be written as $a+b\sqrt{5}$ for some integers $a, b$.
The (field) norm $N$ of an element $a+b\sqrt{5}$ is […]

Equivalent Conditions For a Prime Ideal in a Commutative Ring
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 $J \subset P$.
Proof. […]

A Rational Root of a Monic Polynomial with Integer Coefficients is an Integer
Suppose that $\alpha$ is a rational root of a monic polynomial $f(x)$ in $\Z[x]$.
Prove that $\alpha$ is an integer.
Proof.
Suppose that $\alpha=\frac{p}{q}$ is a rational number in lowest terms, that is, $p$ and $q$ are relatively prime […]

Group of Order $pq$ is Either Abelian or the Center is Trivial
Let $G$ be a group of order $|G|=pq$, where $p$ and $q$ are (not necessarily distinct) prime numbers.
Then show that $G$ is either abelian group or the center $Z(G)=1$.
Hint.
Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […]

Is a Set of All Nilpotent Matrix a Vector Space?
Let $V$ denote the vector space of all real $n\times n$ matrices, where $n$ is a positive integer.
Determine whether the set $U$ of all $n\times n$ nilpotent matrices is a subspace of the vector space $V$ or not.
Definition.
An matrix $A$ is a nilpotent matrix if […]