Homework for REDUCED ROW ECHELON FORM

1. Solving Systems of Linear Equations using RREF

• compute the Reduced Row Echelon Form (RREF) of a matrix $A$
• use RREF$(A)$ to solve a linear system
• compute the rank of a matrix $A$
• classify a linear system by the rank of its augmented matrix

• Examples:
  • From http://linear.ups.edu/html/section-RREF.html , read Example RREF, Example NRREF, Example SAB, Example SAA, Example SAE, and Sage RREF.
    Remember to put your solutions in vector form. For example, the solution to Example SAA in vector form is $${\bf x} \ = \ \left[\begin{array}{cc} 3-r \cr 2+r \cr r \end{array}\right] \ = \ \left[\begin{array}{cc} 3 \cr 2 \cr 0 \end{array}\right] + r \, \left[\begin{array}{cc} -1 \cr 1 \cr 1 \end{array}\right].$$
  • From http://linear.ups.edu/html/section-TSS.html , read Example RREFN, Example ISSI, and Example FDV.
  • The following is a system of six equations in nine variables. Read through until they give the vector form of the solution: http://linear.ups.edu/html/archetype-J.html .
• Homework:
  • From http://linear.ups.edu/html/section-RREF.html , do Exercises RREF.C10, RREF.C11, RREF.C12, RREF.C13, RREF.C14, RREF.C15, RREF.C16, RREF.C17, RREF.C18, RREF.C19, RREF.C30, RREF.C31, RREF.C32, RREF.C33, RREF.M45, and RREF.M50.
    The vector form of the solution for several of the above Exercises is given below:

    RREF.C12	RREF.C14	RREF.C15
    $${\bf x} \ = \ \left[\begin{array}{cc} 2 \cr 0 \cr -6 \cr 0 \end{array}\right] + r \, \left[\begin{array}{cc} -2 \cr 1 \cr 0 \cr 0 \end{array}\right] + s \, \left[\begin{array}{cc} 4 \cr 0 \cr -3 \cr 1 \end{array}\right]$$	$${\bf x} \ = \ \left[\begin{array}{cc} 3 \cr -2 \cr 0 \cr 0 \end{array}\right] + r \, \left[\begin{array}{cc} -2 \cr -3 \cr 1 \cr 0 \end{array}\right] + s \, \left[\begin{array}{cc} -1 \cr 4 \cr 0 \cr 1 \end{array}\right]$$	$${\bf x} \ = \ \left[\begin{array}{cc} 3 \cr -5 \cr 7 \cr 0 \end{array}\right] + r \, \left[\begin{array}{cc} -2 \cr 3 \cr -4 \cr 1 \end{array}\right]$$

  • From http://linear.ups.edu/html/section-TSS.html , do Exercises TSS.C21, TSS.C22, TSS.C23, TSS.C24, TSS.C25, TSS.C26, TSS.C27, and TSS.C28.
    Remember to put your solutions in vector form.
  • From http://linear.ups.edu/html/section-VO.html , do Exercise VO.C15.
  • From http://linear.ups.edu/html/section-MO.html , do Exercise MO.C14.
• Use the following example and create your own problems:

2. Solving Homogeneous Systems of Linear Equations using RREF

• identify a homogeneous system
• use RREF$(A)$ to solve a homogeneous system
• Note: A homogeneous system is ALWAYS consistent

• Examples:
  • From http://linear.ups.edu/html/section-HSE.html , read Example HUSAB, Example HISAA, and Example HISAD.
• Homework:
  • From http://linear.ups.edu/html/section-HSE.html , do Exercises HSE.C21, HSE.C22, HSE.C23, HSE.C25, HSE.C26, and HSE.C27.
    Remember to put your solutions in vector form.