More from my site

• Is the Following Function $T:\R^2 \to \R^3$ a Linear Transformation? Determine whether the function $T:\R^2 \to \R^3$ defined by \[T\left(\, \begin{bmatrix} x \\ y \end{bmatrix} \,\right) = \begin{bmatrix} x_+y \\ x+1 \\ 3y \end{bmatrix}\] is a linear transformation.   Solution. The […]
• Use Cramer’s Rule to Solve a $2\times 2$ System of Linear Equations Use Cramer's rule to solve the system of linear equations \begin{align*} 3x_1-2x_2&=5\\ 7x_1+4x_2&=-1. \end{align*}   Solution. Let \[A=[A_1, A_2]=\begin{bmatrix} 3 & -2\\ 7& 4 \end{bmatrix},\] be the coefficient matrix of the system, where $A_1, A_2$ […]
• Prove that a Group of Order 217 is Cyclic and Find the Number of Generators Let $G$ be a finite group of order $217$. (a) Prove that $G$ is a cyclic group. (b) Determine the number of generators of the group $G$.     Sylow's Theorem We will use Sylow's theorem to prove part (a). For a review of Sylow's theorem, check out the […]
• Even Perfect Numbers and Mersenne Prime Numbers Prove that if $2^n-1$ is a Mersenne prime number, then \[N=2^{n-1}(2^n-1)\] is a perfect number. On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.   Definitions. In this post, a […]
• Group Homomorphism from $\Z/n\Z$ to $\Z/m\Z$ When $m$ Divides $n$ Let $m$ and $n$ be positive integers such that $m \mid n$. (a) Prove that the map $\phi:\Zmod{n} \to \Zmod{m}$ sending $a+n\Z$ to $a+m\Z$ for any $a\in \Z$ is well-defined. (b) Prove that $\phi$ is a group homomorphism. (c) Prove that $\phi$ is surjective. (d) Determine […]
• A ring is Local if and only if the set of Non-Units is an Ideal 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 […]
• Matrix Operations with Transpose Calculate the following expressions, using the following matrices: \[A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}\] (a) $A B^\trans + \mathbf{v} […]
• The Transpose of a Nonsingular Matrix is Nonsingular Let $A$ be an $n\times n$ nonsingular matrix. Prove that the transpose matrix $A^{\trans}$ is also nonsingular.   Definition (Nonsingular Matrix). By definition, $A^{\trans}$ is a nonsingular matrix if the only solution to […]