Homework 3

This HW is due on 09th October 2024. Please write down the solutions neatly in your own handwriting, and submit the completed HW by email by the beginning of class on 09th October 2024. Please submit the HW as a single pdf file - you can scan / take a photo of your completed HW (handwritten) and convert it into a pdf file (please name your file first.last_hw3.pdf). To get credit, please provide all details and give complete reasoning for all your work.

Exercise 1 (2 Points) If a set of vectors are mutually orthogonal, are they always linearly independent? Justify.

Exercise 2 (2 Points) If a set of vectors are orthonormal, are they always linearly independent? Justify.

Exercise 3 (2 Points) If a vector is orthogonal with itself, what can you say about it?

Exercise 4 (2 Points) Show that if a square matrix \mathbf{Q} has orthonormal columns, then \mathbf{Q}^T\mathbf{Q}=\mathbf{I}. What if \mathbf{Q} is not square?

Exercise 5 (2 Points) Let W be the column space (a subspace of \mathbb{R}^3) of the matrix given below. Find a basis for its orthogonal complement W^\perp. \begin{bmatrix} 1& 2& -1 & 3 \\ -2& 0& 2 & 1\\ -1& 2& 1 & 4 \end{bmatrix}