Let $V$ denote the vector space of all real $2\times 2$ matrices.
Suppose that the linear transformation from $V$ to $V$ is given as below.
\[T(A)=\begin{bmatrix}
2 & 3\\
5 & 7
\end{bmatrix}A-A\begin{bmatrix}
2 & 3\\
5 & 7
\end{bmatrix}.\]
Prove or disprove that the linear transformation $T:V\to V$ is an isomorphism.
Add to solve later A square matrix $A$ is called idempotent if $A^2=A$.
(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.
Prove that $P$ is an idempotent matrix.
(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be unit vectors in $\R^n$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
Let $Q=\mathbf{u}\mathbf{u}^{\trans}+\mathbf{v}\mathbf{v}^{\trans}$.
Prove that $Q$ is an idempotent matrix.
(c) Prove that each nonzero vector of the form $a\mathbf{u}+b\mathbf{v}$ for some $a, b\in \R$ is an eigenvector corresponding to the eigenvalue $1$ for the matrix $Q$ in part (b).
Add to solve later A ring is called local if it has a unique maximal ideal.
(a) Prove that a ring $R$ with $1$ is local if and only if the set of non-unit elements of $R$ is an ideal of $R$.
(b) Let $R$ be a ring with $1$ and suppose that $M$ is a maximal ideal of $R$.
Prove that if every element of $1+M$ is a unit, then $R$ is a local ring.
Add to solve later Let
\[R=\left\{\, \begin{bmatrix}
a & b\\
0& a
\end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\]
Then the usual matrix addition and multiplication make $R$ an ring.
Let
\[J=\left\{\, \begin{bmatrix}
0 & b\\
0& 0
\end{bmatrix} \quad \middle | \quad b \in \Q \,\right\}\]
be a subset of the ring $R$.
(a) Prove that the subset $J$ is an ideal of the ring $R$.
(b) Prove that the quotient ring $R/J$ is isomorphic to $\Q$.
Add to solve later Let $R$ be the ring of all $2\times 2$ matrices with integer coefficients:
\[R=\left\{\, \begin{bmatrix}
a & b\\
c& d
\end{bmatrix} \quad \middle| \quad a, b, c, d\in \Z \,\right\}.\]
Let $S$ be the subset of $R$ given by
\[S=\left\{\, \begin{bmatrix}
s & 0\\
0& s
\end{bmatrix} \quad \middle | \quad s\in \Z \,\right\}.\]
(a) True or False: $S$ is a subring of $R$.
(b) True or False: $S$ is an ideal of $R$.
Add to solve later Let $G$ be a nilpotent group and let $H$ be a proper subgroup of $G$.
Then prove that $H \subsetneq N_G(H)$, where $N_G(H)$ is the normalizer of $H$ in $G$.
Add to solve later Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is,
\[H=\{ a\in G \mid \text{the order of $a$ is finite}\}.\]
Prove that $H$ is a subgroup of $G$.
Add to solve later Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.
Add to solve later Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).
Add to solve later Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD).
Add to solve later Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$.
Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$.
Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.
Add to solve later Prove the following statements.
(a) If $a\neq 1$ is an idempotent element of $R$, then $a$ is a zero divisor.
(b) Suppose that $R$ is an integral domain. Determine all the idempotent elements of $R$.
Add to solve later It’s been a year since I started this math blog!!
More than 500 problems were posted during a year (July 19th 2016-July 19th 2017).
I made a list of the 10 math problems on this blog that have the most views.
Can you solve all of them?
The level of difficulty among the top 10 problems.
【★★★】 Difficult (Final Exam Level)
【★★☆】 Standard(Midterm Exam Level)
【★☆☆】 Easy (Homework Level)
Read solution
Add to solve later Prove that a positive definite matrix has a unique positive definite square root.
Add to solve later Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$.
Find all the square roots of the matrix
\[A=\begin{bmatrix}
2 & 2\\
2& 2
\end{bmatrix}.\]
Add to solve later (a) Prove that the matrix $A=\begin{bmatrix}
0 & 1\\
0& 0
\end{bmatrix}$ does not have a square root.
Namely, show that there is no complex matrix $B$ such that $B^2=A$.
(b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root matrices.
Add to solve later Let $F$ be a finite field.
Prove that each element in the field $F$ is the sum of two squares in $F$.
Add to solve later Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers.
Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups.
Add to solve later Using the numbers appearing in
\[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\]
we construct the matrix \[A=\begin{bmatrix}
3 & 14 &1592& 65358\\
97932& 38462643& 38& 32\\
7950& 2& 8841& 9716\\
939937510& 5820& 974& 9
\end{bmatrix}.\]
Prove that the matrix $A$ is nonsingular.
Add to solve later 
