# Pythagorean triple 2017

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• 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 […]
• 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 […]
• Simple Commutative Relation on Matrices Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that $A(A+B)^{-1}B=B(A+B)^{-1}A.$ (University of California, Berkeley Qualifying Exam) Proof. Let $P=A+B$. Then $B=P-A$. Using these, we express the given […]
• Prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$ Using the Matrix Components Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector. Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$.   Solution. We will use the matrix components $A = (a_{i j})_{1 \leq i, j \leq n}$, $B = […] • Order of Product of Two Elements in a Group Let$G$be a group. Let$a$and$b$be elements of$G$. If the order of$a, b$are$m, n$respectively, then is it true that the order of the product$ab$divides$mn$? If so give a proof. If not, give a counterexample. Proof. We claim that it is not true. As a […] • Abelian Normal Subgroup, Intersection, and Product of Groups Let$G$be a group and let$A$be an abelian subgroup of$G$with$A \triangleleft G$. (That is,$A$is a normal subgroup of$G$.) If$B$is any subgroup of$G$, then show that $A \cap B \triangleleft AB.$ Proof. First of all, since$A \triangleleft G$, the […] • Subset of Vectors Perpendicular to Two Vectors is a Subspace Let$\mathbf{a}$and$\mathbf{b}$be fixed vectors in$\R^3$, and let$W$be the subset of$\R^3$defined by $W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.$ Prove that the subset$W\$ is a subspace of […]
• Determinant of a General Circulant Matrix Let \[A=\begin{bmatrix} a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\ a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\ a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\ \vdots & \vdots & \dots & \vdots & \vdots \\ a_{2} & a_3 & \dots & a_{0} & a_{1}\\ a_{1} & a_2 & […]