More from my site

• Example of an Element in the Product of Ideals that Cannot be Written as the Product of Two Elements Let $I=(x, 2)$ and $J=(x, 3)$ be ideal in the ring $\Z[x]$. (a) Prove that $IJ=(x, 6)$. (b) Prove that the element $x\in IJ$ cannot be written as $x=f(x)g(x)$, where $f(x)\in I$ and $g(x)\in J$.   Hint. If $I=(a_1,\dots, a_m)$ and $J=(b_1, \dots, b_n)$ are […]
• 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 […]
• Linearity of Expectations E(X+Y) = E(X) + E(Y) Let $X, Y$ be discrete random variables. Prove the linearity of expectations described as \[E(X+Y) = E(X) + E(Y).\] Solution. The joint probability mass function of the discrete random variables $X$ and $Y$ is defined by \[p(x, y) = P(X=x, Y=y).\] Note that the […]
• True or False. Every Diagonalizable Matrix is Invertible Is every diagonalizable matrix invertible?   Solution. The answer is No. Counterexample We give a counterexample. Consider the $2\times 2$ zero matrix. The zero matrix is a diagonal matrix, and thus it is diagonalizable. However, the zero matrix is not […]
• 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 […]
• 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 […]
• Find the Conditional Probability About Math Exam Experiment A researcher conducted the following experiment. Students were grouped into two groups. The students in the first group had more than 6 hours of sleep and took a math exam. The students in the second group had less than 6 hours of sleep and took the same math exam. The pass […]
• Are These Linear Transformations? Define two functions $T:\R^{2}\to\R^{2}$ and $S:\R^{2}\to\R^{2}$ by \[ T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x+y \\ 0 \end{bmatrix} ,\; S\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} x+y […]