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• Find All the Eigenvalues of 4 by 4 Matrix Find all the eigenvalues of the matrix \[A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 &0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.\] (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. We compute the […]
• Beautiful Formulas for pi=3.14… The number $\pi$ is defined a s the ratio of a circle's circumference $C$ to its diameter $d$: \[\pi=\frac{C}{d}.\] $\pi$ in decimal starts with 3.14... and never end. I will show you several beautiful formulas for $\pi$.   Art Museum of formulas for $\pi$ […]
• Exponential Functions are Linearly Independent Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers. Show that exponential functions \[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\] are linearly independent over $\R$. Hint. Consider a linear combination \[a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.\] […]
• Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix Consider the following system of linear equations \begin{align*} 2x+3y+z&=-1\\ 3x+3y+z&=1\\ 2x+4y+z&=-2. \end{align*} (a) Find the coefficient matrix $A$ for this system. (b) Find the inverse matrix of the coefficient matrix found in (a) (c) Solve the system using […]
• Quotient Group of Abelian Group is Abelian Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$. Then prove that the quotient group $G/N$ is also an abelian group.   Proof. Each element of $G/N$ is a coset $aN$ for some $a\in G$. Let $aN, bN$ be arbitrary elements of $G/N$, where $a, b\in […]
• A Group of Order $20$ is Solvable Prove that a group of order $20$ is solvable.   Hint. Show that a group of order $20$ has a unique normal $5$-Sylow subgroup by Sylow's theorem. See the post summary of Sylow’s Theorem to review Sylow's theorem. Proof. Let $G$ be a group of order $20$. The […]
• Is the Determinant of a Matrix Additive? Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$. Is it true that \[\det(A+B)=\det(A)+\det(B)?\] If so, then give a proof. If not, then give a counterexample.   Solution. We claim that the statement is false. As a counterexample, […]
• Linearity of Expectations E(X+Y) = E(X) + E(Y) Let $X, Y$ be discrete random variables. Prove the linearity of expectations described as \[E(X+Y) = E(X) + E(Y).\] Solution. The joint probability mass function of the discrete random variables $X$ and $Y$ is defined by \[p(x, y) = P(X=x, Y=y).\] Note that the […]