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  • Conjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the SetConjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the Set Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$. For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$.   Proof. $(\subset)$ We first show that $gC_G(X)g^{-1} \subset C_G(gXg^{-1})$. Take any $h\in C_G(X)$. Then for […]
  • A Relation between the Dot Product and the TraceA Relation between the Dot Product and the Trace Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors. Prove that $\tr ( \mathbf{v} \mathbf{w}^\trans ) = \mathbf{v}^\trans \mathbf{w}$.   Solution. Suppose the vectors have components \[\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n […]
  • Union of Two Subgroups is Not a GroupUnion of Two Subgroups is Not a Group Let $G$ be a group and let $H_1, H_2$ be subgroups of $G$ such that $H_1 \not \subset H_2$ and $H_2 \not \subset H_1$. (a) Prove that the union $H_1 \cup H_2$ is never a subgroup in $G$. (b) Prove that a group cannot be written as the union of two proper […]
  • Cosine and Sine Functions are Linearly IndependentCosine and Sine Functions are Linearly Independent Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$. Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.   Proof. Note that the zero vector in the vector space $C[-\pi, \pi]$ is […]
  • In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable. Show that the matrix $A=\begin{bmatrix} 1 & \alpha\\ 0& 1 \end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$. Comment. Remark that if $A$ is a square […]
  • Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly IndependentUsing the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent By calculating the Wronskian, determine whether the set of exponential functions \[\{e^x, e^{2x}, e^{3x}\}\] is linearly independent on the interval $[-1, 1]$.   Solution. The Wronskian for the set $\{e^x, e^{2x}, e^{3x}\}$ is given […]
  • Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of SystemsDetermine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems Determine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations. (a) $\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 6 \end{array} \right]$. (b) […]
  • Ring Homomorphisms and Radical IdealsRing Homomorphisms and Radical Ideals Let $R$ and $R'$ be commutative rings and let $f:R\to R'$ be a ring homomorphism. Let $I$ and $I'$ be ideals of $R$ and $R'$, respectively. (a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that $\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$ (c) Suppose that $f$ is […]

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