# math-magic

by Yu ·

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### More from my site

- Possibilities of the Number of Solutions of a Homogeneous System of Linear Equations Here is a very short true or false problem. Select either True or False. Then click "Finish quiz" button. You will be able to see an explanation of the solution by clicking "View questions" button.
- If the Order of a Group is Even, then the Number of Elements of Order 2 is Odd Prove that if $G$ is a finite group of even order, then the number of elements of $G$ of order $2$ is odd. Proof. First observe that for $g\in G$, \[g^2=e \iff g=g^{-1},\] where $e$ is the identity element of $G$. Thus, the identity element $e$ and the […]
- Matrix Representation of a Linear Transformation of Subspace of Sequences Satisfying Recurrence Relation Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ for $k=1, 2, \dots$. (a) […]
- Injective Group Homomorphism that does not have Inverse Homomorphism Let $A=B=\Z$ be the additive group of integers. Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$. (a) Prove that $\phi$ is a group homomorphism. (b) Prove that $\phi$ is injective. (c) Prove that there does not exist a group homomorphism $\psi:B […]
- Galois Group of the Polynomial $x^p-2$. Let $p \in \Z$ be a prime number. Then describe the elements of the Galois group of the polynomial $x^p-2$. Solution. The roots of the polynomial $x^p-2$ are \[ \sqrt[p]{2}\zeta^k, k=0,1, \dots, p-1\] where $\sqrt[p]{2}$ is a real $p$-th root of $2$ and $\zeta$ […]
- If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian? Let $R$ be a commutative ring with $1$. Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$. Is it true that $A$ is also a Noetherian ring? Proof. The answer is no. We give a counterexample. Let […]
- The Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not Isomorphic Prove that the rings $\Z[x]$ and $\Q[x]$ are not isomoprhic. Proof. We give three proofs. The first two proofs use only the properties of ring homomorphism. The third proof resort to the units of rings. If you are familiar with units of $\Z[x]$, then the […]
- Linear 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 […]