Consider the Initial Value Problem: $$ \begin{array}{111} x_{1}^{\prime} & =6 x_{1}+3 x_{2} & x_{1}(0) & =8 x_{2}^{\prime} & =-15 x_{1}-6 x_{2}, & x_{2}(0) & = 7 \end{array} $$ (a) Find the eigenvalues and eigenvectors for the coefficient matrix. $$ \lambda_{1}=\square, "overrightarrow{\boldsymbol{v}}_{1}=\left[\begin{array}{1} \square 1 \square \end{array} ight], \text { and } \lambda_{2}=\square, \overrightarrow{\boldsymbol{v}}_{\mathbf{2}}=\left[\begin{array}{1} \square \square \end{array} ight] $$ (b) Solve the initial value problem. Give your solution in real form. $$ \begin{array}{1} X_{1}=\square X_{2}=\square \end{array} $$ An ellipse with clockwise orientation 1. Use the phase plotter pplane9.m in MATLAB to describe the trajectory. CS. JG. 130 Introduce the data set with columns the repeated observations of data. $$ A=\left(\begin{array}{ccc} 1 & 2 & 1.5 W 2 & 3 & 3.4 4 & 4 & 6.2 5 & 6.1 & 7.3 \end{array} ight) $$ a) What is the singular decomposition of $A$ as $U D V^{t}$. b) What is the closest rank one matrix to the data given by the columns of $A$ ? That is, perform data reduction on $A$. Of course 4 by 3 matrix CS. JG. 131