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Suppose $A$ is a $2 \times 2$ real matrix with an eigenvalue $\lambda = 2 + 4 i$ and corresponding eigenvector

$\boldsymbol{\vec{v}} = \left\lbrack \begin{array}{c} -1+i \\ i \end{array} \right\rbrack.$
Determine a fundamental set (i.e., linearly independent set) of solutions for $\boldsymbol{\vec{y}^{\,\prime}} = A \boldsymbol{\vec{y}}$ , where the fundamental set consists entirely of real solutions.

Enter your solutions below. Use $t$ as the independent variable in your answers.

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

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

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

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