Group-Theory2

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• Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$. Let $A \in V$ and consider the set $S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}$ of $n^2$ […]
• Transpose of a Matrix and Eigenvalues and Related Questions Let $A$ be an $n \times n$ real matrix. Prove the followings. (a) The matrix $AA^{\trans}$ is a symmetric matrix. (b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal. (c) The matrix $AA^{\trans}$ is non-negative definite. (An $n\times n$ […]
• Quiz 2. The Vector Form For the General Solution / Transpose Matrices. Math 2568 Spring 2017. (a) The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $\left[\begin{array}{rrrrr|r} 1 & 0 & -1 & 0 &-2 & 0 \\ 0 & 1 & 2 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ \end{array} \right].$ […]
• A Prime Ideal in the Ring $\Z[\sqrt{10}]$ Consider the ring $\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}$ and its ideal $P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.$ Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.   Definition of a prime ideal. An ideal $P$ of a ring $R$ is […]
• The Rank of the Sum of Two Matrices Let $A$ and $B$ be $m\times n$ matrices. Prove that $\rk(A+B) \leq \rk(A)+\rk(B).$ Proof. Let $A=[\mathbf{a}_1, \dots, \mathbf{a}_n] \text{ and } B=[\mathbf{b}_1, \dots, \mathbf{b}_n],$ where $\mathbf{a}_i$ and $\mathbf{b}_i$ are column vectors of $A$ and $B$, […]
• A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero Let $U$ and $V$ be vector spaces over a scalar field $\F$. Let $T: U \to V$ be a linear transformation. Prove that $T$ is injective (one-to-one) if and only if the nullity of $T$ is zero.   Definition (Injective, One-to-One Linear Transformation). A linear […]
• Prove that any Algebraic Closed Field is Infinite Prove that any algebraic closed field is infinite.     Definition. A field $F$ is said to be algebraically closed if each non-constant polynomial in $F[x]$ has a root in $F$. Proof. Let $F$ be a finite field and consider the polynomial \[f(x)=1+\prod_{a\in […]
• Torsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian Group Let $A$ be an abelian group and let $T(A)$ denote the set of elements of $A$ that have finite order. (a) Prove that $T(A)$ is a subgroup of $A$. (The subgroup $T(A)$ is called the torsion subgroup of the abelian group $A$ and elements of $T(A)$ are called torsion […]