Question

The matrix $A = \begin{bmatrix} -6 & 6 \\ -\frac{3}{2} & -6 \end{bmatrix}$ has the following eigenvalues and eigenvectors: $\lambda_1 = -6 + 3i$ with $\vec{v}_1 = \begin{bmatrix} -2i \\ 1 \end{bmatrix}$. and $\lambda_2 = -6 - 3i$ with $\vec{v}_2 = \begin{bmatrix} 2i \\ 1 \end{bmatrix}$. Consider the system $x_1' = -6x_1 + 6x_2$ $x_2' = -\frac{3}{2}x_1 - 6x_2$ 1. Determine a fundamental set (i.e., linearly independent set) of solutions for the system, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use $t$ as the independent variable in your answers. $\vec{x}^{(1)}(t) = \begin{bmatrix} \\ \end{bmatrix}$ and $\vec{x}^{(2)}(t) = \begin{bmatrix} \\ \end{bmatrix}$ 2. Write the general real solution for the linear system (write "c1" and "c2" for $c_1$ and $c_2$) $x_1(t) =$ $x_2(t) =$

Name: the matrix a beginbmatrix 6 6 frac32 6 endbmatrix has the following eigenvalues and eigenvectors lambda1 6 3i with vecv1 beginbmatrix 2i 1 endbmatrix and lambda2 6 3i with vecv2 beginbma 60237
Uploaded: 2024-06-15T13:27:26-08:00
Duration: 3 min 47 s
Channel: Shu Naito
Description: the matrix a beginbmatrix 6 6 frac32 6 endbmatrix has the following eigenvalues and eigenvectors lambda1 6 3i with vecv1 beginbmatrix 2i 1 endbmatrix and lambda2 6 3i with vecv2 beginbma 60237

          The matrix $A = \begin{bmatrix} -6 & 6 \\ -\frac{3}{2} & -6 \end{bmatrix}$ has the following eigenvalues and eigenvectors:

$\lambda_1 = -6 + 3i$ with $\vec{v}_1 = \begin{bmatrix} -2i \\ 1 \end{bmatrix}$.

and

$\lambda_2 = -6 - 3i$ with $\vec{v}_2 = \begin{bmatrix} 2i \\ 1 \end{bmatrix}$.

Consider the system

$x_1' = -6x_1 + 6x_2$
$x_2' = -\frac{3}{2}x_1 - 6x_2$

1. Determine a fundamental set (i.e., linearly independent set) of solutions for the system, where the fundamental set consists entirely of real solutions.

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

$\vec{x}^{(1)}(t) = \begin{bmatrix}  \\   \end{bmatrix}$ and $\vec{x}^{(2)}(t) = \begin{bmatrix}  \\   \end{bmatrix}$

2. Write the general real solution for the linear system (write "c1" and "c2" for $c_1$ and $c_2$)

$x_1(t) =$
$x_2(t) =$

$the matrix a beginbmatrix 6 6 frac32 6 endbmatrix has the following eigenvalues and eigenvectors lambda1 6 3i with vecv1 beginbmatrix 2i 1 endbmatrix and lambda2 6 3i with vecv2 beginbma 60237$

Added by Lawrence B.

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