# Purdue-univeristy-exam-eye-catch

• Ascending Chain of Submodules and Union of its Submodules Let $R$ be a ring with $1$. Let $M$ be an $R$-module. Consider an ascending chain $N_1 \subset N_2 \subset \cdots$ of submodules of $M$. Prove that the union $\cup_{i=1}^{\infty} N_i$ is a submodule of $M$.   Proof. To simplify the notation, let us […]
• Subspaces of the Vector Space of All Real Valued Function on the Interval Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not. (a) $S=\{f(x) \in V \mid f(0)=f(1)\}$. (b) $T=\{f(x) \in V \mid […] • Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let$G$be a group of order$12$. Prove that$G$has a normal subgroup of order$3$or$4$. Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow$p$-subgroup in a group$GH$, then it is […] • Compute Determinant of a Matrix Using Linearly Independent Vectors Let$A$be a$3 \times 3$matrix. Let$\mathbf{x}, \mathbf{y}, \mathbf{z}$are linearly independent$3$-dimensional vectors. Suppose that we have $A\mathbf{x}=\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, A\mathbf{y}=\begin{bmatrix} 0 \\ 1 \\ 0 […] • 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 […] • Matrix Operations with Transpose Calculate the following expressions, using the following matrices: \[A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$ (a)$A B^\trans + \mathbf{v} […]
• If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$ Let $N$ be a normal subgroup of a group $G$. Suppose that $G/N$ is an infinite cyclic group. Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.   Hint. Use the fourth (or Lattice) isomorphism theorem. Proof. […]
• 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 […]