Mathematics About the Number 2018 Happy New Year 2018!!
Here are several mathematical facts about the number 2018.
Is 2018 a Prime Number?
The number 2018 is an even number, so in particular 2018 is not a prime number.
The prime factorization of 2018 is
\[2018=2\cdot 1009.\]
Here $2$ and $1009$ are […]

Welcome to Problems in Mathematics
Welcome to my website.
I post problems and their solutions/proofs in mathematics.
Most of the problems are undergraduate level mathematics.
Here are several topics I cover on this website.
Topics
Linear Algebra
Group Theory
Ring Theory
Field Theory, Galois Theory
Module […]

Mathematics About the Number 2017 Happy New Year 2017!!
Here is the list of mathematical facts about the number 2017 that you can brag about to your friends or family as a math geek.
2017 is a prime number
Of course, I start with the fact that the number 2017 is a prime number.
The previous prime year was […]

Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector
Let
\[A=\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix}
1\\
0
\end{bmatrix}.\]
Compute $A^{2017}\mathbf{u}$.
(The Ohio State University, Linear Algebra Exam)
Solution.
We first compute $A\mathbf{u}$. We […]

Companion Matrix for a Polynomial
Consider a polynomial
\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.
Define the matrix
\[A=\begin{bmatrix}
0 & 0 & \dots & 0 &-a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
0 & 1 & \dots & 0 & -a_2 \\
\vdots & […]

Nilpotent Matrix and Eigenvalues of the Matrix
An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix.
Prove the followings.
(a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero.
(b) The matrix $A$ is nilpotent if and only if […]

Powers of a Diagonal Matrix
Let $A=\begin{bmatrix}
a & 0\\
0& b
\end{bmatrix}$.
Show that
(1) $A^n=\begin{bmatrix}
a^n & 0\\
0& b^n
\end{bmatrix}$ for any $n \in \N$.
(2) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix.
Show that $B^n=S^{-1}A^n S$ for any $n \in […]

Prove Vector Space Properties Using Vector Space Axioms
Using the axiom of a vector space, prove the following properties.
Let $V$ be a vector space over $\R$. Let $u, v, w\in V$.
(a) If $u+v=u+w$, then $v=w$.
(b) If $v+u=w+u$, then $v=w$.
(c) The zero vector $\mathbf{0}$ is unique.
(d) For each $v\in V$, the additive inverse […]