# Math-Magic Tree example

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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 $7, 5, 3, 2, […] • Show that the Given 2 by 2 Matrix is Singular Consider the matrix$M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$. (a) Show that$M$is singular. (b) Find a non-zero vector$\mathbf{v}$such that$M \mathbf{v} = \mathbf{0}$, where$\mathbf{0}$is the$2$-dimensional zero vector. Solution. (a) Show […] • Is the Derivative Linear Transformation Diagonalizable? Let$\mathrm{P}_2$denote the vector space of polynomials of degree$2$or less, and let$T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$be the derivative linear transformation, defined by $T( ax^2 + bx + c ) = 2ax + b .$ Is$T$diagonalizable? If so, find a diagonal matrix which […] • Using Gram-Schmidt Orthogonalization, Find an Orthogonal Basis for the Span Using Gram-Schmidt orthogonalization, find an orthogonal basis for the span of the vectors$\mathbf{w}_{1},\mathbf{w}_{2}\in\R^{3}$if $\mathbf{w}_{1} = \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} ,\quad \mathbf{w}_{2} = \begin{bmatrix} 2 \\ -1 \\ […] • Find All the Square Roots of a Given 2 by 2 Matrix Let A be a square matrix. A matrix B satisfying B^2=A is call a square root of A. Find all the square roots of the matrix \[A=\begin{bmatrix} 2 & 2\\ 2& 2 \end{bmatrix}.$ Proof. Diagonalize$A$. We first diagonalize the matrix […] • Find a Basis for Nullspace, Row Space, and Range of a Matrix Let$A=\begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 &3 & 0 & 5 \\ 1 & 1 & 6 & 3 \end{bmatrix}$. (a) Find a basis for the nullspace of$A$. (b) Find a basis for the row space of$A$. (c) Find a basis for the range of$A$that consists of column vectors of$A$. (d) […] • The Quadratic Integer Ring$\Z[\sqrt{5}]$is not a Unique Factorization Domain (UFD) Prove that the quadratic integer ring$\Z[\sqrt{5}]$is not a Unique Factorization Domain (UFD). Proof. Every element of the ring$\Z[\sqrt{5}]$can be written as$a+b\sqrt{5}$for some integers$a, b$. The (field) norm$N$of an element$a+b\sqrt{5}\$ is […]
• 10 True or False Problems about Basic Matrix Operations Test your understanding of basic properties of matrix operations. There are 10 True or False Quiz Problems. These 10 problems are very common and essential. So make sure to understand these and don't lose a point if any of these is your exam problems. (These are actual exam […]