Consider the linear system

$$\displaystyle \boldsymbol{ \vec{y}^{\,\prime} } = \left\lbrack \begin{array}{rr} -3 & -2 \\ 6 & 4 \end{array} \right\rbrack \boldsymbol{\vec{y}}.$$

1. Find the eigenvalues and eigenvectors for the coefficient matrix.

   $\lambda_1 =$

   , $\boldsymbol{\vec{v}_1} =$

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

   , and $\lambda_2 =$

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

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

   
   
   
2. For each eigenpair in the previous part, form a solution of $\boldsymbol{\vec{y}^{\,\prime}} = A \boldsymbol{\vec{y}}$. Use $t$ as the independent variable in your answers.

   $\boldsymbol{\vec{y}_1}(t) =$

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

   and $\boldsymbol{\vec{y}_2}(t) =$

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

   
   
   
3. Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solutions? Choose Yes, it is a fundamental set No, it is not a fundamental set