More from my site

• 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 […]
• Companion Matrix for a Polynomial Consider a polynomial \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Define the matrix \[A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & […]
• If a Group $G$ Satisfies $abc=cba$ then $G$ is an Abelian Group Let $G$ be a group with identity element $e$. Suppose that for any non identity elements $a, b, c$ of $G$ we have \[abc=cba. \tag{*}\] Then prove that $G$ is an abelian group.   Proof. To show that $G$ is an abelian group we need to show that \[ab=ba\] for any […]
• Intersection of Two Null Spaces is Contained in Null Space of Sum of Two Matrices Let $A$ and $B$ be $n\times n$ matrices. Then prove that \[\calN(A)\cap \calN(B) \subset \calN(A+B),\] where $\calN(A)$ is the null space (kernel) of the matrix $A$.   Definition. Recall that the null space (or kernel) of an $n \times n$ matrix […]
• Solve a Linear Recurrence Relation Using Vector Space Technique Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be a subspace of $V$ defined by \[U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.\] Let $T$ be the linear transformation from […]
• Find a Nonsingular Matrix Satisfying Some Relation Determine whether there exists a nonsingular matrix $A$ if \[A^2=AB+2A,\] where $B$ is the following matrix. If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$. (a) \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 1 & 2 & […]
• Ring is a Filed if and only if the Zero Ideal is a Maximal Ideal Let $R$ be a commutative ring. Then prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal of $R$.   Proof. $(\implies)$: If $R$ is a field, then $\{0\}$ is a maximal ideal Suppose that $R$ is a field and let $I$ be a non zero ideal: \[ \{0\} […]
• A Condition that a Vector is a Linear Combination of Columns Vectors of a Matrix Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$. Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the […]