#### Linear Systems

5. Consider the system x' following values of c: (a) c = (0,00) 9 x where c is a parameter. Classify the geometry and stability properties of the system for the (b) c = 0 (c) c = (-1,0) (d) c= -1 (e) c = (-∞, -1)

2.a. Express the function below in terms of unit step function. Calculate its derivative and provide a sketch of it. b. Express the function below in terms of unit step function. Sketch its integral.

1. Determine whether the following signals are energy signals, power signals or neither.Clearly show proof.

5. Which of the following systems are time-varying or time invariant. Show proof a. y(t) = x(t− 3) b.y(t) = x(t) cos(wt) C.d^2y/dt² + 2ty(t) = x(t)

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a. find the eigenvalues (real or complex) of, b.what are the eigenvalues of (M-121)?

Consider the following linear program: \begin{array}{l} \max x_{1}+x_{2} \\ \text { s.t.: } \\ \text { Cl } 3 x_{1}+2 x_{2} \leq 12 \\ \text { C2 } 2 x_{1}+3 x_{2} \leq 12 \\ \text { C3 } 2 x_{1}+2 x_{2} \leq 9 \\ \text { C4 } 2 x_{1}+2 x_{2} \geq 3 \\ \text { CS } \quad x_{1}, x_{2} \geq 0 \end{array} a) (5 points) Graph the feasible region of the LP. Is the feasible region unbounded? b) (35 points) Solve this problem using Simplex algorithm. Make sure to indicate: * Whether or not you need to perform Phase * Show the BV and NBV for each iteration of Simplex * Show the steps of the algorithm in the graph from point a) * Is this a unique or multiple solution?

4. Determine which of the following systems is linear or non-linear. Show proof a. y(t) = 2x(t) – 5 b.·dy/dt + 4y(t) = 2x(t) C.dy/dt+2ty(t) = x(t) d.dy/dt+ 4y(t) + 1 = x(t) e.(dy/dt) + y(t) = x²(t) f. (dy/dt)² + y(t) = x(t)

3. Consider the system tx' = [22]x, t> 0 with initial condition X(2) = [−12]. A Assuming solutions of the form x = where X, V are an eigenvalue/eigenvector pair of the given matrix, use techniques similar to those used to construct solutions to the constant coefficient linear homogeneous systems to solve the given initial value problem. Write your answer as a single vector.