# Math-Magic Tree empty

by Yu · Published · Updated

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- 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, […]
- True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample. Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$. If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a […]
- Normalize Lengths to Obtain an Orthonormal Basis Let \[ \mathbf{v}_{1} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} ,\; \mathbf{v}_{2} = \begin{bmatrix} 1 \\ -1 \end{bmatrix} . \] Let $V=\Span(\mathbf{v}_{1},\mathbf{v}_{2})$. Do $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ form an orthonormal basis for $V$? If […]
- If a Group is of Odd Order, then Any Nonidentity Element is Not Conjugate to its Inverse Let $G$ be a finite group of odd order. Assume that $x \in G$ is not the identity element. Show that $x$ is not conjugate to $x^{-1}$. Proof. Assume the contrary, that is, assume that there exists $g \in G$ such that $gx^{-1}g^{-1}=x$. Then we have \[xg=gx^{-1}. […]
- Find Matrix Representation of Linear Transformation From $\R^2$ to $\R^2$ Let $T: \R^2 \to \R^2$ be a linear transformation such that \[T\left(\, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 4 \\ 1 \end{bmatrix}, T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 3 \\ 2 […]
- Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix Let $A$ be an $n\times n$ matrix with the characteristic polynomial \[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable. (a) Find the size of the matrix $A$. (b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue […]
- Complex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues Let $A$ be an $n\times n$ real matrix. Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$. We give two proofs. Proof 1. Let $\mathbf{x}$ be an eigenvector corresponding to the […]
- Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.) Solution. For example, let $A$ be the following $3\times […]