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• Determine the Values of $a$ such that the 2 by 2 Matrix is Diagonalizable Let \[A=\begin{bmatrix} 1-a & a\\ -a& 1+a \end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is a complex number. Determine the values of $a$ such that the matrix $A$ is diagonalizable. (Nagoya University, Linear Algebra Exam Problem)   Proof. To […]
• Irreducible Polynomial $x^3+9x+6$ and Inverse Element in Field Extension Prove that the polynomial \[f(x)=x^3+9x+6\] is irreducible over the field of rational numbers $\Q$. Let $\theta$ be a root of $f(x)$. Then find the inverse of $1+\theta$ in the field $\Q(\theta)$.   Proof. Note that $f(x)$ is a monic polynomial and the prime […]
• Find a Formula for a Linear Transformation If $L:\R^2 \to \R^3$ is a linear transformation such that \begin{align*} L\left( \begin{bmatrix} 1 \\ 0 \end{bmatrix}\right) =\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \,\,\,\, L\left( \begin{bmatrix} 1 \\ 1 \end{bmatrix}\right) =\begin{bmatrix} 2 \\ 3 […]
• $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain. Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.   Proof. Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$. Define the function $\Psi:R[x,y] \to R[t]$ sending […]
• 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 […]
• $\sqrt[m]{2}$ is an Irrational Number Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.   Hint. Use ring theory: Consider the polynomial $f(x)=x^m-2$. Apply Eisenstein's criterion, show that $f(x)$ is irreducible over $\Q$. Proof. Consider the monic polynomial […]
• Independent Events of Playing Cards A card is chosen randomly from a deck of the standard 52 playing cards. Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart. Prove or disprove that the events $E$ and $F$ are independent. Definition of Independence Events […]
• Express a Vector as a Linear Combination of Other Vectors Express the vector $\mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 \end{bmatrix}$ as a linear combination of the vectors \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2= \begin{bmatrix} 1 \\ 2 \\ 1 […]