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(50 points) Consider the linear system \[ \overrightarrow{\boldsymbol{y}}^{\prime}=\left[\begin{array}{rr} -3 & -2 \\ 5 & 3 \end{array}\right] \overrightarrow{\boldsymbol{y}} \] a. Find the eigenvalues and eigenvectors for the coefficient matrix. \[ \lambda_{1}=\quad \vec{v}_{1}=\left[\quad, \quad \text { and } \lambda_{2}=\left[\vec{v}_{2}=[\right.\right. \] b. Find the real-valued solution to the initial value problem \[ \left\{\begin{array}{lll} y_{1}^{\prime}=-3 y_{1}-2 y_{2}, & y_{1}(0)=-1 \\ y_{2}^{\prime}= & 5 y_{1}+3 y_{2}, & y_{2}(0)=5 \end{array}\right. \] Use \( t \) as the independent variable in your answers. \[ \begin{array}{l} y_{1}(t)= \\ y_{2}(t)= \end{array} \]

Given system of equations is :- Y?=[?3?253]Y Its characteristic equation is :- |A??I|=0 |?3???253??|=0?(?3??)(3??)?(?2)×5=0 ??9+3??3?+?2+10=0 ??2+1=0?