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  • If the Quotient by the Center is Cyclic, then the Group is AbelianIf the Quotient by the Center is Cyclic, then the Group is Abelian Let $Z(G)$ be the center of a group $G$. Show that if $G/Z(G)$ is a cyclic group, then $G$ is abelian. Steps. Write $G/Z(G)=\langle \bar{g} \rangle$ for some $g \in G$. Any element $x\in G$ can be written as $x=g^a z$ for some $z \in Z(G)$ and $a \in \Z$. Using […]
  • The Intersection of Two Subspaces is also a SubspaceThe Intersection of Two Subspaces is also a Subspace Let $U$ and $V$ be subspaces of the $n$-dimensional vector space $\R^n$. Prove that the intersection $U\cap V$ is also a subspace of $\R^n$.   Definition (Intersection). Recall that the intersection $U\cap V$ is the set of elements that are both elements of $U$ […]
  • No/Infinitely Many Square Roots of 2 by 2 MatricesNo/Infinitely Many Square Roots of 2 by 2 Matrices (a) Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ does not have a square root. Namely, show that there is no complex matrix $B$ such that $B^2=A$. (b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root […]
  • The Determinant of a Skew-Symmetric Matrix is ZeroThe Determinant of a Skew-Symmetric Matrix is Zero Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd.   Definition (Skew-Symmetric) A matrix $A$ is called skew-symmetric if $A^{\trans}=-A$. Here $A^{\trans}$ is the transpose of $A$. Proof. Properties of […]
  • Linear Independent Continuous FunctionsLinear 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 […]
  • Find Values of $a$ so that the Matrix is NonsingularFind Values of $a$ so that the Matrix is Nonsingular Let $A$ be the following $3 \times 3$ matrix. \[A=\begin{bmatrix} 1 & 1 & -1 \\ 0 &1 &2 \\ 1 & 1 & a \end{bmatrix}.\] Determine the values of $a$ so that the matrix $A$ is nonsingular.   Solution. We use the fact that a matrix is nonsingular if and only if […]
  • Matrices Satisfying $HF-FH=-2F$Matrices Satisfying $HF-FH=-2F$ Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation \[HF-FH=-2F.\] (a) Find the trace of the matrix $F$. (b) Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ […]
  • Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian GroupsSurjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism. Prove that we have an isomorphism of groups: \[G \cong \ker(f)\times \Z.\]   Proof. Since $f:G\to \Z$ is surjective, there exists an element $a\in G$ such […]

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