# Pythagorean triple 2017

• 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 […]
• There is Exactly One Ring Homomorphism From the Ring of Integers to Any Ring Let $\Z$ be the ring of integers and let $R$ be a ring with unity. Determine all the ring homomorphisms from $\Z$ to $R$.   Definition. Recall that if $A, B$ are rings with unity then a ring homomorphism $f: A \to B$ is a map […]
• Group Homomorphism Sends the Inverse Element to the Inverse Element Let $G, G'$ be groups. Let $\phi:G\to G'$ be a group homomorphism. Then prove that for any element $g\in G$, we have $\phi(g^{-1})=\phi(g)^{-1}.$     Definition (Group homomorphism). A map $\phi:G\to G'$ is called a group homomorphism […]
• Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices Let $A, B, C$ be the following $3\times 3$ matrices. $A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & […] • Linear Algebra Midterm 1 at the Ohio State University (3/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 3 and contains […] • Find Matrix Representation of Linear Transformation From \R^2 to \R^2 Let T: \R^2 \to \R^2 be a linear transformation such that \[T\left(\, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 4 \\ 1 \end{bmatrix}, T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 3 \\ 2 […] • A Matrix Having One Positive Eigenvalue and One Negative Eigenvalue 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)   Solution. Let us put […]
• Nilpotent Matrices and Non-Singularity of Such Matrices Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix. Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity […]