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Perfect Numbers and Mersenne Prime Numbers


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  • The Order of a Conjugacy Class Divides the Order of the GroupThe Order of a Conjugacy Class Divides the Order of the Group Let $G$ be a finite group. The centralizer of an element $a$ of $G$ is defined to be \[C_G(a)=\{g\in G \mid ga=ag\}.\] A conjugacy class is a set of the form \[\Cl(a)=\{bab^{-1} \mid b\in G\}\] for some $a\in G$. (a) Prove that the centralizer of an element of $a$ […]
  • Nontrivial Action of a Simple Group on a Finite SetNontrivial Action of a Simple Group on a Finite Set Let $G$ be a simple group and let $X$ be a finite set. Suppose $G$ acts nontrivially on $X$. That is, there exist $g\in G$ and $x \in X$ such that $g\cdot x \neq x$. Then show that $G$ is a finite group and the order of $G$ divides $|X|!$. Proof. Since $G$ acts on $X$, it […]
  • Complement of Independent Events are IndependentComplement of Independent Events are Independent Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$. Prove that $E$ and $F^c$ are independent as well. Solution. Note that $E\cap F$ and $E \cap F^c$ are disjoint and $E = (E \cap F) \cup (E \cap F^c)$. It follows that \[P(E) = P(E \cap F) + P(E […]
  • The Sum of Cosine Squared in an Inner Product SpaceThe Sum of Cosine Squared in an Inner Product Space Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$. Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$. Prove that \[\cos […]
  • Properties of Nonsingular and Singular MatricesProperties of Nonsingular and Singular Matrices An $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$. Otherwise $A$ is called singular. (a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is […]
  • Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or LessFind 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) […]
  • Quiz 3. Condition that Vectors are Linearly Dependent/ Orthogonal Vectors are Linearly IndependentQuiz 3. Condition that Vectors are Linearly Dependent/ Orthogonal Vectors are Linearly Independent (a) For what value(s) of $a$ is the following set $S$ linearly dependent? \[ S=\left \{\,\begin{bmatrix} 1 \\ 2 \\ 3 \\ a \end{bmatrix}, \begin{bmatrix} a \\ 0 \\ -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ a^2 […]
  • Diagonalizable by an Orthogonal Matrix Implies a Symmetric MatrixDiagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.   Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]

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