# Diagonalization-eye-catch

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• The set of $2\times 2$ Symmetric Matrices is a Subspace Let $V$ be the vector space over $\R$ of all real $2\times 2$ matrices. Let $W$ be the subset of $V$ consisting of all symmetric matrices. (a) Prove that $W$ is a subspace of $V$. (b) Find a basis of $W$. (c) Determine the dimension of $W$.   Proof. […]
• Non-Abelian Simple Group is Equal to its Commutator Subgroup Let $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.   Definitions/Hint. We first recall relevant definitions. A group is called simple if its normal subgroups are either the trivial subgroup or the group […]
• Linear Independent Continuous Functions Let $C[3, 10]$ be the vector space consisting of all continuous functions defined on the interval $[3, 10]$. Consider the set $S=\{ \sqrt{x}, x^2 \}$ in $C[3,10]$. Show that the set $S$ is linearly independent in $C[3,10]$.   Proof. Note that the zero vector […]
• Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ Let $A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix} 4 \\ -1 \\ -7 \end{bmatrix}.$ Find $A^{10}\mathbf{v}$. You may use the following information without proving […]
• The Order of a Conjugacy Class Divides the Order of the Group Let $G$ be a finite group. The centralizer of an element $a$ of $G$ is defined to be $C_G(a)=\{g\in G \mid ga=ag\}.$ A conjugacy class is a set of the form $\Cl(a)=\{bab^{-1} \mid b\in G\}$ for some $a\in G$. (a) Prove that the centralizer of an element of $a$ […]
• The Inverse Matrix is Unique Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.   Hint. That the inverse matrix of $A$ is unique means that there is only one inverse matrix of $A$. (That's why we say "the" inverse matrix of $A$ and denote it by […]
• Find a Matrix that Maps Given Vectors to Given Vectors Suppose that a real matrix $A$ maps each of the following vectors $\mathbf{x}_1=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ into the […]
• Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors Find a basis for $\Span(S)$ where $S= \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} , \begin{bmatrix} -1 \\ -2 \\ -1 \end{bmatrix} , \begin{bmatrix} 2 \\ 6 \\ -2 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} \right\}$.   Solution. We […]