# math-magic

by Yu ·

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### More from my site

- Non-Prime Ideal of Continuous Functions Let $R$ be the ring of all continuous functions on the interval $[0,1]$. Let $I$ be the set of functions $f(x)$ in $R$ such that $f(1/2)=f(1/3)=0$. Show that the set $I$ is an ideal of $R$ but is not a prime ideal. Proof. We first show that $I$ is an ideal of […]
- If There are 28 Elements of Order 5, How Many Subgroups of Order 5? Let $G$ be a group. Suppose that the number of elements in $G$ of order $5$ is $28$. Determine the number of distinct subgroups of $G$ of order $5$. Solution. Let $g$ be an element in $G$ of order $5$. Then the subgroup $\langle g \rangle$ generated by $g$ is […]
- If the Matrix Product $AB=0$, then is $BA=0$ as Well? Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix. Is it true that the matrix product with opposite order $BA$ is also the zero matrix? If so, give a proof. If not, give a […]
- Conjugate 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 […]
- Is the Following Function $T:\R^2 \to \R^3$ a Linear Transformation? Determine whether the function $T:\R^2 \to \R^3$ defined by \[T\left(\, \begin{bmatrix} x \\ y \end{bmatrix} \,\right) = \begin{bmatrix} x_+y \\ x+1 \\ 3y \end{bmatrix}\] is a linear transformation. Solution. The […]
- Equation $x_1^2+\cdots +x_k^2=-1$ Doesn’t Have a Solution in Number Field $\Q(\sqrt[3]{2}e^{2\pi i/3})$ Let $\alpha= \sqrt[3]{2}e^{2\pi i/3}$. Prove that $x_1^2+\cdots +x_k^2=-1$ has no solutions with all $x_i\in \Q(\alpha)$ and $k\geq 1$. Proof. Note that $\alpha= \sqrt[3]{2}e^{2\pi i/3}$ is a root of the polynomial $x^3-2$. The polynomial $x^3-2$ is […]
- Two Quotients Groups are Abelian then Intersection Quotient is Abelian Let $K, N$ be normal subgroups of a group $G$. Suppose that the quotient groups $G/K$ and $G/N$ are both abelian groups. Then show that the group \[G/(K \cap N)\] is also an abelian group. Hint. We use the following fact to prove the problem. Lemma: For a […]
- Prove that the Center of Matrices is a Subspace Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define \[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\] The set $W$ is called the center of $V$. Prove that $W$ is a subspace […]