More from my site

• If a Subgroup $H$ is in the Center of a Group $G$ and $G/H$ is Nilpotent, then $G$ is Nilpotent Let $G$ be a nilpotent group and let $H$ be a subgroup such that $H$ is a subgroup in the center $Z(G)$ of $G$. Suppose that the quotient $G/H$ is nilpotent. Then show that $G$ is also nilpotent.   Definition (Nilpotent Group) We recall here the definition of a […]
• Subset of Vectors Perpendicular to Two Vectors is a Subspace Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by \[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\] Prove that the subset $W$ is a subspace of […]
• Projection to the subspace spanned by a vector Let $T: \R^3 \to \R^3$ be the linear transformation given by orthogonal projection to the line spanned by $\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}$. (a) Find a formula for $T(\mathbf{x})$ for $\mathbf{x}\in \R^3$. (b) Find a basis for the image subspace of $T$. (c) Find […]
• Linear Independent Continuous Functions Let $C[3, 10]$ be the vector space consisting of all continuous functions defined on the interval $[3, 10]$. Consider the set \[S=\{ \sqrt{x}, x^2 \}\] in $C[3,10]$. Show that the set $S$ is linearly independent in $C[3,10]$.   Proof. Note that the zero vector […]
• 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 < […]
• The Center of a p-Group is Not Trivial Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$. (Such a group is called a $p$-group.) Show that the center $Z(G)$ of the group $G$ is not trivial.   Hint. Use the class equation. Proof. If $G=Z(G)$, then the statement is true. So suppose that $G\neq […]
• No Nonzero Zero Divisor in a Field / Direct Product of Rings is Not a Field (a) Let $F$ be a field. Show that $F$ does not have a nonzero zero divisor. (b) Let $R$ and $S$ be nonzero rings with identities. Prove that the direct product $R\times S$ cannot be a field.   Proof. (a) Show that $F$ does not have a nonzero zero divisor. […]
• Probability that Alice Wins n Games Before Bob Wins m Games Alice and Bob play some game against each other. The probability that Alice wins one game is $p$. Assume that each game is independent. If Alice wins $n$ games before Bob wins $m$ games, then Alice becomes the champion of the game. What is the probability that Alice becomes the […]