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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 $\boldsymbol{\vec{v}_2} =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$		$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

Find the solution to the linear system of differential equations $\displaystyle \left\lbrace \begin{array}{rcl} x^{\,\prime} & = & -6x-7y \\ y^{\,\prime} & = & 14x+15y \end{array} \right.$ satisfying the initial conditions $x(0) = 4$ and $y(0) = -6$ .

$x(t) =$
$y(t) =$

You can earn partial credit on this problem.