# normal-subgroup

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• Find a General Formula of a Linear Transformation From $\R^2$ to $\R^3$ Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying $T\left(\, \begin{bmatrix} 1 \\ 2 \end{bmatrix}\,\right)=\begin{bmatrix} 3 \\ 4 \\ 5 \end{bmatrix} \text{ and } T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} […] • All the Eigenvectors of a Matrix Are Eigenvectors of Another Matrix Let A and B be an n \times n matrices. Suppose that all the eigenvalues of A are distinct and the matrices A and B commute, that is AB=BA. Then prove that each eigenvector of A is an eigenvector of B. (It could be that each eigenvector is an eigenvector for […] • Calculate Determinants of Matrices Calculate the determinants of the following n\times n matrices. \[A=\begin{bmatrix} 1 & 0 & 0 & \dots & 0 & 0 &1 \\ 1 & 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 1 & 1 & \dots & 0 & 0 & 0 \\ \vdots & \vdots […] • 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 $\pi$ […]
• Summary: Possibilities for the Solution Set of a System of Linear Equations In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems. Determine all possibilities for the solution set of the system of linear equations described below. (a) A homogeneous system of $3$ […]
• Perturbation of a Singular Matrix is Nonsingular Suppose that $A$ is an $n\times n$ singular matrix. Prove that for sufficiently small $\epsilon>0$, the matrix $A-\epsilon I$ is nonsingular, where $I$ is the $n \times n$ identity matrix. Hint. Consider the characteristic polynomial $p(t)$ of the matrix $A$. Note […]
• 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 […]
• 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. […]