More from my site

• The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers. Decipher the trick of the following mathematical magic.   The Rule of the Game Here is the game. Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below. For example, let's say I have chosen the numbers $7, 5, 3, 2, […]
• True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample. Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$. If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a […]
• Condition that a Function Be a Probability Density Function Let $c$ be a positive real number. Suppose that $X$ is a continuous random variable whose probability density function is given by \begin{align*} f(x) = \begin{cases} \frac{1}{x^3} & \text{ if } x \geq c\\ 0 & \text{ if } x < […]
• Determine Whether Each Set is a Basis for $\R^3$ Determine whether each of the following sets is a basis for $\R^3$. (a) $S=\left\{\, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \\ 4 \end{bmatrix} […]
• Idempotent (Projective) Matrices are Diagonalizable Let $A$ be an $n\times n$ idempotent complex matrix. Then prove that $A$ is diagonalizable.   Definition. An $n\times n$ matrix $A$ is said to be idempotent if $A^2=A$. It is also called projective matrix. Proof. In general, an $n \times n$ matrix $B$ is […]
• The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD) Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).   Proof. Every element of the ring $\Z[\sqrt{5}]$ can be written as $a+b\sqrt{5}$ for some integers $a, b$. The (field) norm $N$ of an element $a+b\sqrt{5}$ is […]
• Find All the Square Roots of a Given 2 by 2 Matrix Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$. Find all the square roots of the matrix \[A=\begin{bmatrix} 2 & 2\\ 2& 2 \end{bmatrix}.\]   Proof. Diagonalize $A$. We first diagonalize the matrix […]
• Find All Symmetric Matrices satisfying the Equation Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$? Express your solution using free variable(s).   Solution. Let $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ […]