# Boston-college-exam-eye-catch

by Yu ·

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- Prove the Cauchy-Schwarz Inequality Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$. Prove the Cauchy-Schwarz inequality: \[|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.\] We give two proofs. Proof 1 Let $x$ be a variable and consider the length of the vector […]
- Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8 Prove that every group of order $24$ has a normal subgroup of order $4$ or $8$. Proof. Let $G$ be a group of order $24$. Note that $24=2^3\cdot 3$. Let $P$ be a Sylow $2$-subgroup of $G$. Then $|P|=8$. Consider the action of the group $G$ on […]
- 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 […]
- Solve the System of Linear Equations and Give the Vector Form for the General Solution Solve the following system of linear equations and give the vector form for the general solution. \begin{align*} x_1 -x_3 -2x_5&=1 \\ x_2+3x_3-x_5 &=2 \\ 2x_1 -2x_3 +x_4 -3x_5 &= 0 \end{align*} (The Ohio State University, linear algebra midterm exam […]
- Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials Let $A$ be a $2\times 2$ real symmetric matrix. Prove that all the eigenvalues of $A$ are real numbers by considering the characteristic polynomial of $A$. Proof. Let $A=\begin{bmatrix} a& b \\ c& d \end{bmatrix}$. Then […]
- Equivalent Definitions of Characteristic Subgroups. Center is Characteristic. Let $H$ be a subgroup of a group $G$. We call $H$ characteristic in $G$ if for any automorphism $\sigma\in \Aut(G)$ of $G$, we have $\sigma(H)=H$. (a) Prove that if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$. (b) Prove that the center […]
- Multiplicative Groups of Real Numbers and Complex Numbers are not Isomorphic Let $\R^{\times}=\R\setminus \{0\}$ be the multiplicative group of real numbers. Let $\C^{\times}=\C\setminus \{0\}$ be the multiplicative group of complex numbers. Then show that $\R^{\times}$ and $\C^{\times}$ are not isomorphic as groups. Recall. Let $G$ and $K$ […]
- The Product of a Subgroup and a Normal Subgroup is a Subgroup Let $G$ be a group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. The product of $H$ and $N$ is defined to be the subset \[H\cdot N=\{hn\in G\mid h \in H, n\in N\}.\] Prove that the product $H\cdot N$ is a subgroup of […]