# perfect-numbers

• The 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 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 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 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 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 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 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 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 […]