Project 2

Watch the relevant videos and do the following:

1. Create the following matrix (using QQ for coefficients)

(1)
\begin{pmatrix} 1 & 2 & 2 & 4\\ 1 & 3 & 3 & 5\\ 2 & 6 & 5 & 6 \\ \end{pmatrix}

Then, follow the procedures from class (outlined in 1.4) to find the reduced row echelon form for this matrix using only the three elementary row operations described in the video. You can check your answer using the .rref() command.

2. Find three different echelon forms for the matrix $\begin{pmatrix}2& 1 &7 &-7& 2\\ -3& 4& -5 &-6 & 3\\ 1& 1& 4 &-5 &2\end{pmatrix}$.

3. Use the process of Gaussian Elimination (showing each step) to find the reduced echelon form for the matrix

(2)
\begin{align} \begin{pmatrix}1 &0 &-3&0&6&0&7&-5& 9\\ 0&0& 0& 5&0&1&0& 3& -7\\ 0&0& 0& 0&0&0&0& 0& 0\\ 0&1& 0& 0&0&0&0&-4 &2\\0& 0& 0& 0& 0& 0& 1& 7& 3\\ 0&0&0&0&0&0&0&0&0\end{pmatrix}. \end{align}
  • Check that the previous answer is correct by using the built in functionality of Sage to compute the echelon form (note: use .rref() appended to a matrix for reduced row echelon form.)

4. Consider the system of (three) linear equations: $2x_1 +x_2 +7x_3 -7x_4=8;\ \ -3x_1 +4x_2 -5x_3 -6x_4=−12;\ \ x_1 +x_2 +4x_3 -5x_4=4$

  • Use the built in "solve" function to solve this system.
  • Use the echelon form of an appropriate matrix to solve this system.
  • Are the answers the same? Which method do you prefer to use? Why?
  • Write the general solution to the system in standard form.

5. For the matrix $\begin{pmatrix}2&3&5\\ -1& 4& 6\\ 3& 10& 2\\ 3& -1 &-1\\ 6&9&3\end{pmatrix}$,

  • Find the rank using the echelon form.
  • Find the rank using the built in function.
  • Suppose the matrix represents an augmented matrix for a system of linear equations. Is there a solution to this system? Explain.

6. Define four different 4 by 4 matrices (using M=random_matrix(QQ,4) four times ) and find their reduced echelon forms. Did anything interesting happen?

7. Read Chapter 1.6.1 on curve fitting in the book. Then complete problem 19 from 1.6 using Sage.