Math Insight

Quiz 1

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Total points: 1
  1. Let $\vc{x} = (x,y)$ be the vector of state variables for the discrete dynamical system \begin{align*} \vc{x}_{n+1} &= A\vc{x}_n \qquad \text{for $n=0,1,2,\ldots$}\\ \vc{x}_0 &= (-2.9, -1) \end{align*} where \begin{align*} A = \left[\begin{matrix}-0.5 & 1.8\\-3.8 & 3\end{matrix}\right]. \end{align*}

    Simulate this system (using a computer program might be helpful) for 10 time steps. The final value of the state variable vector is $\vc{x}_{10} = $
    .
    Include at least 5 significant digits in your answer.

  2. Consider the following $2$-dimensional system of discrete dynamical equations: \begin{eqnarray*} x_{n+1} &=& - x_{n} - 2 y_{n}\\ y_{n+1} &=& x_{n} + 2 y_{n}\\ x_{0} &=& 7\\ y_{0} &=& 0 \end{eqnarray*}
    1. Compute the next five values of $x$ and $y$.
      $x_1 =$
      ,  $y_1 =$

      $x_2 =$
      ,  $y_2 =$

      $x_3 =$
      ,  $y_3 =$

      $x_4 =$
      ,  $y_4 =$

      $x_5 =$
      ,  $y_5 =$
    2. Convert the system into a matrix equation.

      $\begin{bmatrix} x_{n+1}\\ y_{n+1} \end{bmatrix}=$




      $\begin{bmatrix} x_n\\ y_n\end{bmatrix}$

  3. Solve the matrix equation $A \vc{x} = \vc{b}$, where $$A = \left[\begin{matrix}-5 & -3\\-5 & 4\end{matrix}\right], \quad \vc{b} = \left[\begin{matrix}-12\\-19\end{matrix}\right], \quad \text{and} \quad \vc{x} = \left[\begin{matrix}x\\y\end{matrix}\right].$$ $\vc{x} =$

  4. Compute the following matrix-vector products.
    1. $\left[\begin{matrix}4 & 3\\5 & -5\end{matrix}\right]\left[\begin{matrix}3\\-1\end{matrix}\right]=$
    2. $\left[\begin{matrix}2 & 6\\0 & 0\end{matrix}\right]\left[\begin{matrix}-4\\1\end{matrix}\right]=$

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Math 2241, Spring 2023