# Math-Magic Tree filled

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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, […] • Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) $\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right.$ where$a,b,c, d$[…] • Eigenvalues and Algebraic/Geometric Multiplicities of Matrix$A+cI$Let$A$be an$n \times n$matrix and let$c$be a complex number. (a) For each eigenvalue$\lambda$of$A$, prove that$\lambda+c$is an eigenvalue of the matrix$A+cI$, where$I$is the identity matrix. What can you say about the eigenvectors corresponding to […] • A Linear Transformation$T: U\to V$cannot be Injective if$\dim(U) > \dim(V)$Let$U$and$V$be finite dimensional vector spaces over a scalar field$\F$. Consider a linear transformation$T:U\to V$. Prove that if$\dim(U) > \dim(V)$, then$T$cannot be injective (one-to-one). Hints. You may use the folowing facts. A linear […] • Subspace Spanned By Cosine and Sine Functions Let$\calF[0, 2\pi]$be the vector space of all real valued functions defined on the interval$[0, 2\pi]$. Define the map$f:\R^2 \to \calF[0, 2\pi]$by $\left(\, f\left(\, \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta […] • Compute the Product A^{2017}\mathbf{u} of a Matrix Power and a Vector Let \[A=\begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix} 1\\ 0 \end{bmatrix}.$ Compute$A^{2017}\mathbf{u}$. (The Ohio State University, Linear Algebra Exam) Solution. We first compute$A\mathbf{u}$. We […] • The Null Space (the Kernel) of a Matrix is a Subspace of$\R^n$Let$A$be an$m \times n$real matrix. Then the null space$\calN(A)$of$A$is defined by $\calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.$ That is, the null space is the set of solutions to the homogeneous system$A\mathbf{x}=\mathbf{0}_m$. Prove that the […] • Determine the Values of$a$so that$W_a$is a Subspace For what real values of$a$is the set $W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}$ a subspace of the vector space$C(\mathbb{R})$of all real-valued functions? Solution. The zero element of$C(\mathbb{R})$is the function$\mathbf{0}\$ defined by […]