# HW1

- Due Oct 22, 2016 by 11:59pm
- Points 45
- Submitting a file upload

**EE 590 B (PMP): Numerical Methods for Electrical Engineering Homework 1 **

**Assigned: Thursday, October 13, 2016, Due: Saturday, October 22, 2016 **

**Instructor: Tamara Bonaci **

**Department of Electrical Engineering University of Washington, Seattle **

Your homework should be submitted as PDF files using the course dropbox. They are due by **6pm on Saturday, October 22, 2016****. **

**Problem 1 **

Please find the 1-norms, 2-norms and the infinity norms of the following vectors: 2

1. x_{1} = [2; -3; 1]

2. x_{2 }= [1; 1; 1]

**Problem 2:**

Find the ranks of matrices:

1. A_{1} = [0 1 0; 0 0 0; 0 0 1]

2. A_{2} = [4 1 -1; 3 2 0; 1 1 0]

3. A_{3} = [1 2 3 4; 0 -1 -2 2; 0 0 0 1]

**Problem 3 **

Find the determinants and the traces of the following matrices by hand:

1. A_{1} = [1 1; 5 -3]

2. A_{2} = [2 0 0; 0 3 0; 0 0 0]

**Problem 4 **

Check to see if the following matrices are positive definite, positive semidefinite, negative definite, negative semidefinite or indefinite:

1. A_{1} = [2 3 2; 3 1 0; 2 0 2]

2. A_{2} = [0 0 -1; 0 0 0; -1 0 0]

**Problem 5 **

Please determine if the following signals are periodic. If they are, determine their fundamental period:

1. x(t) = π cos(5t) + 7e^{j2}^{t} − je^{j4}^{t}

2. x(t) = $\sum $ δ(t − k) (the sum goes from - infinity to infinity)

**Problem 6 **

Given input signal *x*(*t*) and output signal *y*(*t*), please show whether the following systems are linear or not.

1. y(t) = cos(x(t))

2. y(t) = x(t) cos(t)

3. y(t) = x(t − 1) + x(t − 2)

4. y(t)= $\int $x(t)dt (the integral goes from 0 to t)

**Problem 7 **

Consider the following continuous-time system:

y(t) = sin(|x(t)|)

Determine if the given system is time invariant. Please explain your answer.

**Problem 8 **

Consider the continuous-time system with input *x*(t) and output *y*(t) related by:

y(t) = x(sin(t))

1. Is the given system linear?

2. Is it time invariant?

3. Is it causal?

4. Is it memoryless?

Please explain your answers.

**Problem 9 **

Consider the following systems:

1. y_{1}(t) = x(t − 2) + x(2 − t)

2. y_{2}(t) = cos(3t)x(t)

3. y_{3}(t)=x(t/3)

Please determine if the given systems are:

• Linear,

• Time-invariant,

• Causal,

• Memoryless,

• Stable.

**Problem 10 (Extra credit) **

The triangle inequality produces an upper bound on a sum, but it also yields the following lower bound on a difference, referred to as the backward triangle inequality:

Prove that inequality (1) holds true.

|∥x∥−∥y∥| ≤ ∥x−y∥ (1)