Description

Ding Zhao
24-677 Special Topics: Linear Control Systems
• Your online version and its timestamp will be used for assessment. The paper version, will only be used for backup purposes.
• We will use Gradescope ( https://www.gradescope.com/) to grade. The link is on the panel of CANVAS. If you are confused about the tool, post your questions on Piazza. • At the start of every exercise you will see topic(s) on what the given question is about and what will you be learning.
• The Gradescope Autograder python script for submitting your theory homework answers will be given to you. The script is named ”hw3 theory.py” and you have to submit it with your answers. Carefully refer to the further instructions given in the script for submission format.
• For the programming assignment, copy and paste, or export your answer script into a python file named ”hw3 script.py”. Follow the further instructions given in the jupyter notebook named ”PHW3 qs.ipynb” for information on submission format of your answers.
• Submit hw3 theory.py, hw3 script.py, hw3 output.npy to Gradescope under ”Programming Assignment 3” and pdf/images to ”Homework 3”.
• We advise you to start with the assignment early. For information about the late days refer to the Syllabus in Canvas.
Exercise 1. Gram-schmidt method
Find the orthonormal basis using Gram-Schmidt orthogonalization for the following vectors in R3 :

Assume v1 as your first vector.
Exercise 2. Orthogonal complement
Find the orthogonal complement of the subspace of R3 spanned by the two vectors
and
Exercise 3. Orthogonal projection
4×1 + 10×2 − 4×3 = −8
1. Find the minimum distance between the given plane and the origin.
2. Find coordinates of the point on the plane closest to the origin.
Exercise 4. Analyzing solution of equations
Consider the linear algebraic equation: Ax = y

It has three equations and two unknowns.
1. Does a solution x exist ?
2. Is the solution unique?
3. Does the solution exist if y .
Exercise 5. Solution of equations and norm.
1. Find the general solution of

2. Find the solution that has the smallest Euclidean norm.
Exercise 6. Rank and Nullity
Find the rank and nullity of the following matrices:
A A A
Exercise 7. Linear operators
Determine which of the following operations on vectors are linear operators:
1. , where x ∈R3
2. , where x ∈R3
3. where f(σ) is a continuous function, and x is in the vector space of continuous functions.
4. , where