University of Washington, Bothell

CSS 485: Introduction to Artificial Neural Networks
Fall 2002
Homework Assignment 2: Data Reduction

Assigned: Wednesday, October 16, 2002
Due: Wednesday, October 23, 2002 (start of class)

For the following exercises, please remember to show your work. You can get the answers from MATLAB and perhaps other sources; I want to see how you get the answer with pencil and paper.

1. Which of the following sets of vectors are independent? Find the dimension of the vector space spanned by each set. If you like, you may verify your answers using the MATLAB rank function.

   1. $$\left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right] \qquad\left... ... \end{array}\right] \qquad\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right]$$

      
      

   2. $$\left[\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right] \qquad\left... ... \end{array}\right] \qquad\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right]$$

      
      

   3. For extra credit, do the same with the vectors $\sin t$, $\cos t$, and $\cos (2t)$.

2. Textbook, exercise 4.1.

3. Find the eigenvectors and eigenvalues of the following matrices:

   1. $$\left[\begin{array}{rrr} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{array}\right]$$

      
      

   2. $$\left[\begin{array}{rrr} 1 & 2 & 2 \\ 0 & 2 & 1 \\ -1 & 2 & 2 \end{array}\right]$$

      
      

   3. $$\left[\begin{array}{rrr} 1 & 1 & -2 \\ -1 & 2 & 1 \\ 0 & 1 & -1 \end{array}\right]$$

      
      

4. Say that we know that a certain linear transformation $A: \IR^2 \rightarrow \IR^2$ has eigenvalues and eigenvectors given by:

   $$\lambda_1 = 2 \qquad \mathbf{v}_1 = \left[\begin{array}{c}1 \\ 2\... ...= 1 \qquad \mathbf{v}_1 = \left[\begin{array}{c}1 \\ 1\end{array}\right] \qquad$$

   
   
   Find the matrix representation $A$ of the transformation. (You might use the MATLAB function eig to check your answers.)

5. Textbook, exercise 4.8.