# Math-Magic Tree example

by Yu ·

Add to solve later

Add to solve later

Add to solve later

### More from my site

- 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, […]
- Example of an Infinite Group Whose Elements Have Finite Orders Is it possible that each element of an infinite group has a finite order? If so, give an example. Otherwise, prove the non-existence of such a group. Solution. We give an example of a group of infinite order each of whose elements has a finite order. Consider […]
- Eigenvalues of Squared Matrix and Upper Triangular Matrix Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix. If \[P^{-1}AP=\begin{bmatrix} 1 & 2 & 3 \\ 0 &4 &5 \\ 0 & 0 & 6 \end{bmatrix},\] then find all the eigenvalues of the matrix $A^2$. We give two proofs. The first version is a […]
- Solve the System of Linear Equations and Give the Vector Form for the General Solution Solve the following system of linear equations and give the vector form for the general solution. \begin{align*} x_1 -x_3 -2x_5&=1 \\ x_2+3x_3-x_5 &=2 \\ 2x_1 -2x_3 +x_4 -3x_5 &= 0 \end{align*} (The Ohio State University, linear algebra midterm exam […]
- Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$ Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix. Definition (Symmetric Matrix). A matrix $A$ is called symmetric if $A^{\trans}=A$. In terms of entries, an $n\times n$ matrix $A=(a_{ij})$ is […]
- Determine whether the Matrix is Nonsingular from the Given Relation Let $A$ and $B$ be $3\times 3$ matrices and let $C=A-2B$. If \[A\begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}=B\begin{bmatrix} 2 \\ 6 \\ 10 \end{bmatrix},\] then is the matrix $C$ nonsingular? If so, prove it. Otherwise, explain why not. […]
- Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$ Let $V$ be a vector space over the field of real numbers $\R$. Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$. Proof. Since $V$ is an $n$-dimensional vector space, it has a basis \[B=\{\mathbf{v}_1, \dots, […]
- 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 […]