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• Finite Order Matrix and its Trace Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that (a) $|\tr(A)|\leq n$. (b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$. (c) $\tr(A)=n$ if and only if $A=I_n$. Proof. (a) […]
• Ring of Gaussian Integers and Determine its Unit Elements Denote by $i$ the square root of $-1$. Let $R=\Z[i]=\{a+ib \mid a, b \in \Z \}$ be the ring of Gaussian integers. We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to $N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.$ Here $\bar{\alpha}$ is the complex conjugate of […]
• Find Values of $h$ so that the Given Vectors are Linearly Independent Find the value(s) of $h$ for which the following set of vectors $\left \{ \mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} h \\ 1 \\ -h \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2h \\ 3h+1 […] • Inner Product, Norm, and Orthogonal Vectors Let \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3 are vectors in \R^n. Suppose that vectors \mathbf{u}_1, \mathbf{u}_2 are orthogonal and the norm of \mathbf{u}_2 is 4 and \mathbf{u}_2^{\trans}\mathbf{u}_3=7. Find the value of the real number a in […] • Determine the Quotient Ring \Z[\sqrt{10}]/(2, \sqrt{10}) Let \[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}$ be an ideal of the ring $\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.$ Then determine the quotient ring $\Z[\sqrt{10}]/P$. Is $P$ a prime ideal? Is $P$ a maximal ideal?   Solution. We […]
• 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 […]
• Eigenvalues of a Matrix and its Transpose are the Same Let $A$ be a square matrix. Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.   Proof. Recall that the eigenvalues of a matrix are roots of its characteristic polynomial. Hence if the matrices $A$ and $A^{\trans}$ […]
• Characteristic Polynomials of $AB$ and $BA$ are the Same Let $A$ and $B$ be $n \times n$ matrices. Prove that the characteristic polynomials for the matrices $AB$ and $BA$ are the same. Hint. Consider the case when the matrix $A$ is invertible. Even if $A$ is not invertible, note that $A-\epsilon I$ is invertible matrix […]