# Diagonalization-eye-catch

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• Prove that the Dot Product is Commutative: $\mathbf{v}\cdot \mathbf{w}= \mathbf{w} \cdot \mathbf{v}$ Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors. (a) Prove that $\mathbf{v}^\trans \mathbf{w} = \mathbf{w}^\trans \mathbf{v}$. (b) Provide an example to show that $\mathbf{v} \mathbf{w}^\trans$ is not always equal to $\mathbf{w} […] • Dihedral Group and Rotation of the Plane Let$n$be a positive integer. Let$D_{2n}$be the dihedral group of order$2n$. Using the generators and the relations, the dihedral group$D_{2n}$is given by $D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.$ Put$\theta=2 \pi/n$. (a) Prove that the matrix […] • Describe the Range of the Matrix Using the Definition of the Range Using the definition of the range of a matrix, describe the range of the matrix $A=\begin{bmatrix} 2 & 4 & 1 & -5 \\ 1 &2 & 1 & -2 \\ 1 & 2 & 0 & -3 \end{bmatrix}.$ Solution. By definition, the range$\calR(A)$of the matrix$A$is given […] • Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center$Z(G)$Let$G$be a finite group of order$231=3\cdot 7 \cdot 11$. Prove that every Sylow$11$-subgroup of$G$is contained in the center$Z(G)$. Hint. Prove that there is a unique Sylow$11$-subgroup of$G$, and consider the action of$G$on the Sylow$11$-subgroup by […] • Non-Prime Ideal of Continuous Functions Let$R$be the ring of all continuous functions on the interval$[0,1]$. Let$I$be the set of functions$f(x)$in$R$such that$f(1/2)=f(1/3)=0$. Show that the set$I$is an ideal of$R$but is not a prime ideal. Proof. We first show that$I$is an ideal of […] • Basic Properties of Characteristic Groups Definition (automorphism). An isomorphism from a group$G$to itself is called an automorphism of$G$. The set of all automorphism is denoted by$\Aut(G)$. Definition (characteristic subgroup). A subgroup$H$of a group$G$is called characteristic in$G$if for any$\phi […]
• Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$. Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.   Proof. To show that $f(N)$ is normal, we show that $gf(N)g^{-1}=f(N)$ for any $g \in […] • Every Group of Order 72 is Not a Simple Group Prove that every finite group of order$72$is not a simple group. Definition. A group$G$is said to be simple if the only normal subgroups of$G$are the trivial group$\{e\}$or$G$itself. Hint. Let$G$be a group of order$72\$. Use the Sylow's theorem and determine […]