1. The matrix $\displaystyle \left\lbrack \begin{array}{rr} -6 & -7 \\ 14 & 15 \end{array} \right\rbrack$   has eigenvalues $\lambda_1 = 8$ and $\lambda_2 = 1$. Find eigenvectors corresponding to these eigenvalues.
$\boldsymbol{\vec{v}_1} =$
 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$
and $\boldsymbol{\vec{v}_2} =$
 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

2. Find the solution to the linear system of differential equations $\displaystyle \left\lbrack \begin{array}{r} x_1^{\,\prime} \\ x_2^{\,\prime} \end{array} \right\rbrack = \left\lbrack \begin{array}{rr} -6 & -7 \\ 14 & 15 \end{array} \right\rbrack \left\lbrack \begin{array}{r} x_1 \\ x_2 \end{array} \right\rbrack$   satisfying the initial conditions $\displaystyle \left\lbrack \begin{array}{r} x_1(0) \\ x_2(0) \end{array} \right\rbrack = \left\lbrack \begin{array}{r} 4 \\ -6 \end{array} \right\rbrack$.

$x_1(t) =$
$x_2(t) =$