# The mathematical properties of 2017

• 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 […]
• Companion Matrix for a Polynomial Consider a polynomial $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where $a_i$ are real numbers. Define the matrix $A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & […] • Nilpotent Matrix and Eigenvalues of the Matrix An n\times n matrix A is called nilpotent if A^k=O, where O is the n\times n zero matrix. Prove the followings. (a) The matrix A is nilpotent if and only if all the eigenvalues of A is zero. (b) The matrix A is nilpotent if and only if […] • Powers of a Diagonal Matrix Let A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}. Show that (1) A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix} for any n \in \N. (2) Let B=S^{-1}AS, where S be an invertible 2 \times 2 matrix. Show that B^n=S^{-1}A^n S for any n \in […] • Linear Transformation that Maps Each Vector to Its Reflection with Respect to x-Axis Let F:\R^2\to \R^2 be the function that maps each vector in \R^2 to its reflection with respect to x-axis. Determine the formula for the function F and prove that F is a linear transformation. Solution 1. Let \begin{bmatrix} x \\ y […] • Eigenvalues and Eigenvectors of The Cross Product Linear Transformation We fix a nonzero vector \mathbf{a} in \R^3 and define a map T:\R^3\to \R^3 by \[T(\mathbf{v})=\mathbf{a}\times \mathbf{v}$ for all $\mathbf{v}\in \R^3$. Here the right-hand side is the cross product of $\mathbf{a}$ and $\mathbf{v}$. (a) Prove that $T:\R^3\to \R^3$ is […]