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• 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 […]
• Top 10 Popular Math Problems in 2016-2017 It's been a year since I started this math blog!! More than 500 problems were posted during a year (July 19th 2016-July 19th 2017). I made a list of the 10 math problems on this blog that have the most views. Can you solve all of them? The level of difficulty among the top […]
• Cyclic Group if and only if There Exists a Surjective Group Homomorphism From $\Z$ Show that a group $G$ is cyclic if and only if there exists a surjective group homomorphism from the additive group $\Z$ of integers to the group $G$.   Proof. $(\implies)$: If $G$ is cyclic, then there exists a surjective homomorhpism from $\Z$ Suppose that $G$ is […]
• Find a Row-Equivalent Matrix which is in Reduced Row Echelon Form and Determine the Rank For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix. (a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$. (b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$. (c) $C […]
• Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less. Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{align*} (a) […]
• Is the Set of All Orthogonal Matrices a Vector Space? An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$. Let $V$ be the vector space of all real $2\times 2$ matrices. Consider the subset \[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\] Prove or disprove that $W$ is a subspace of […]
• Perturbation of a Singular Matrix is Nonsingular Suppose that $A$ is an $n\times n$ singular matrix. Prove that for sufficiently small $\epsilon>0$, the matrix $A-\epsilon I$ is nonsingular, where $I$ is the $n \times n$ identity matrix. Hint. Consider the characteristic polynomial $p(t)$ of the matrix $A$. Note […]
• Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$ Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$. Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.   Solution. Let $\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}$. Then we want \[\mathbf{v} […]