# Welcome to Problems in Mathematics

Welcome to my website.

I post problems and its solutions/proofs in mathematics almost every day.

Most of the problems are undergraduate level mathematics.

Here are several topics I cover on this website.

2017 is a prime number. Do you want to know more about 2017?

Check out the post Mathematics about the number 2017

For my profile, please check out Who’s Yu Tsumura

• Sylow’s Theorem (Summary) In this post we review Sylow's theorem and as an example we solve the following problem. Show that a group of order $200$ has a normal Sylow $5$-subgroup. Review of Sylow's Theorem One of the important theorems in group theory is Sylow's theorem. Sylow's theorem is a […]
• Fundamental Theorem of Finitely Generated Abelian Groups and its application In this post, we study the Fundamental Theorem of Finitely Generated Abelian Groups, and as an application we solve the following problem. Problem. Let $G$ be a finite abelian group of order $n$. If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic […]
• How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix $A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}$ by finding a nonsingular […]
• 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 […]
• If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$. Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$. Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.   Hint. It follows from […]
• Symmetric Matrix and Its Eigenvalues, Eigenspaces, and Eigenspaces Let $A$ be a $4\times 4$ real symmetric matrix. Suppose that $\mathbf{v}_1=\begin{bmatrix} -1 \\ 2 \\ 0 \\ -1 \end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $1$ of $A$. Suppose that the eigenspace for the eigenvalue $2$ is $3$-dimensional. (a) Find an […]
• Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix 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 […]
• A Square Root Matrix of a Symmetric Matrix Answer the following two questions with justification. (a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix. (b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]