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• Determine the Values of $a$ so that $W_a$ is a Subspace For what real values of $a$ is the set \[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\] a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?   Solution. The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]
• The Range and Null Space of the Zero Transformation of Vector Spaces Let $U$ and $V$ be vector spaces over a scalar field $\F$. Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for each vector $\mathbf{u}\in U$. (a) Prove that $T:U\to V$ is a linear transformation. (Hence, $T$ is called the zero transformation.) (b) Determine […]
• If 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 […]
• Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where \[\mathbf{a}=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix} 1 \\ 1 […]
• Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix Suppose the following information is known about a $3\times 3$ matrix $A$. \[A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=6\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 1 \\ -1 \\ 1 […]
• If a Subgroup Contains a Sylow Subgroup, then the Normalizer is the Subgroup itself Let $G$ be a finite group and $P$ be a nontrivial Sylow subgroup of $G$. Let $H$ be a subgroup of $G$ containing the normalizer $N_G(P)$ of $P$ in $G$. Then show that $N_G(H)=H$.   Hint. Use the conjugate part of the Sylow theorem. See the second statement of the […]
• Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57 Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group. Then determine the number of elements in $G$ of order $3$.   Proof. Observe the prime factorization $57=3\cdot 19$. Let $n_{19}$ be the number of Sylow $19$-subgroups of $G$. By […]
• Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by \[A=I-a\mathbf{v}\mathbf{v}^{\trans},\] where […]