The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers.
Decipher the trick of the following mathematical magic.
The Rule of the Game
Here is the game.
Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below.
For example, let's say I have chosen the numbers $7, 5, 3, 2, […]
If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$
Let $N$ be a normal subgroup of a group $G$.
Suppose that $G/N$ is an infinite cyclic group.
Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.
Hint.
Use the fourth (or Lattice) isomorphism theorem.
Proof. […]
A Linear Transformation Maps the Zero Vector to the Zero Vector
Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively.
Show that $T(\mathbf{0}_n)=\mathbf{0}_m$.
(The Ohio State University Linear Algebra […]
Group Homomorphism, Preimage, and Product of Groups
Let $G, G'$ be groups and let $f:G \to G'$ be a group homomorphism.
Put $N=\ker(f)$. Then show that we have
\[f^{-1}(f(H))=HN.\]
Proof.
$(\subset)$ Take an arbitrary element $g\in f^{-1}(f(H))$. Then we have $f(g)\in f(H)$.
It follows that there exists $h\in H$ […]
Two Quadratic Fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are Not Isomorphic
Prove that the quadratic fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are not isomorphic.
Hint.
Note that any homomorphism between fields over $\Q$ fixes $\Q$ pointwise.
Proof.
Assume that there is an isomorphism $\phi:\Q(\sqrt{2}) \to \Q(\sqrt{3})$.
Let […]
Dihedral Group and Rotation of the Plane
Let $n$ be a positive integer. Let $D_{2n}$ be the dihedral group of order $2n$. Using the generators and the relations, the dihedral group $D_{2n}$ is given by
\[D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.\]
Put $\theta=2 \pi/n$.
(a) Prove that the matrix […]
Elementary Questions about a Matrix
Let
\[A=\begin{bmatrix}
-5 & 0 & 1 & 2 \\
3 &8 & -3 & 7 \\
0 & 11 & 13 & 28
\end{bmatrix}.\]
(a) What is the size of the matrix $A$?
(b) What is the third column of $A$?
(c) Let $a_{ij}$ be the $(i,j)$-entry of $A$. Calculate $a_{23}-a_{31}$.
[…]
Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like.
Consider the matrix
\[A=\begin{bmatrix}
3/2 & 2\\
-1& -3/2
\end{bmatrix} \in M_{2\times 2}(\R).\]
(a) Find the eigenvalues and corresponding eigenvectors of $A$.
(b) Show that for $\mathbf{v}=\begin{bmatrix}
1 \\
0
\end{bmatrix}\in \R^2$, we can choose […]