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  • The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers.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 SubspacesTrue 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 […]
  • Find Values of $a$ so that Augmented Matrix Represents a Consistent SystemFind Values of $a$ so that Augmented Matrix Represents a Consistent System Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations. \[A= \left[\begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 2 &-1 & -2 & a^2 \\ -1 & -7 & -11 & a \end{array} \right],\] where $a$ is a real number. Determine all the […]
  • Eigenvalues and Eigenvectors of The Cross Product Linear TransformationEigenvalues and Eigenvectors of The Cross Product Linear Transformation We fix a nonzero vector $\mathbf{a}$ in $\R^3$ and define a map $T:\R^3\to \R^3$ by \[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}\] for all $\mathbf{v}\in \R^3$. Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$. (a) Prove that $T:\R^3\to \R^3$ is […]
  • The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$ Let $p$ be a prime number. Let $G$ be a non-abelian $p$-group. Show that the index of the center of $G$ is divisible by $p^2$. Proof. Suppose the order of the group $G$ is $p^a$, for some $a \in \Z$. Let $Z(G)$ be the center of $G$. Since $Z(G)$ is a subgroup of $G$, the order […]
  • Vector Space of 2 by 2 Traceless MatricesVector Space of 2 by 2 Traceless Matrices Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers. Let \[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix} a & b\\ c& -a \end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\] (a) Show that $W$ is a subspace of […]
  • Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$ Let \[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\] be an ideal of the ring \[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\] Then determine the quotient ring $\Z[\sqrt{10}]/P$. Is $P$ a prime ideal? Is $P$ a maximal ideal?   Solution. We […]
  • Spanning Sets for $\R^2$ or its SubspacesSpanning Sets for $\R^2$ or its Subspaces In this problem, we use the following vectors in $\R^2$. \[\mathbf{a}=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{b}=\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \mathbf{c}=\begin{bmatrix} 2 \\ 3 \end{bmatrix}, \mathbf{d}=\begin{bmatrix} 3 \\ 2 […]

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