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(Solved):   (50 points) Consider the linear system \[ \overrightarrow{\boldsymbol{y}}^{\prime}=\left[\ ...



(50 points) Consider the linear system
\[
\overrightarrow{\boldsymbol{y}}^{\prime}=\left[\begin{array}{rr}
3 & 2 \\
-6 & -4
\

 

(50 points) Consider the linear system \[ \overrightarrow{\boldsymbol{y}}^{\prime}=\left[\begin{array}{rr} 3 & 2 \\ -6 & -4 \end{array}\right] \overrightarrow{\boldsymbol{y}} . \] a. Find the eigenvalues and eigenvectors for the coefficient matrix. \[ \lambda_{1}=, \overrightarrow{\boldsymbol{v}}_{\mathbf{1}}=\left[, \quad, \text { and } \lambda_{2}=\left[\overrightarrow{\boldsymbol{v}}_{\mathbf{2}}=[]\right.\right. \] b. For each eigenpair in the previous part, form a solution of \( \overrightarrow{\boldsymbol{y}}^{\prime}=A \overrightarrow{\boldsymbol{y}} \). Use \( t \) as the independent variable in your answers. \[ \overrightarrow{\boldsymbol{y}}_{\mathbf{1}}(t)=[\quad] \text { and } \overrightarrow{\boldsymbol{y}}_{\mathbf{2}}(t)=[ \] c. Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solutions?


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Given system of differential equation is y?=[32?6?4]y Let matrix A=[32?6?4] Characteristic equation is |A??I|=0 [3??2?6?4??]=0?(3??)(?4??)+12=0 ?12+4?
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