Homework for Solving Linear Systems

1. Solving Linear Systems

• Compute the Reduced Row Echelon Form (RREF) of a matrix.
• Use RREF to solve a linear system.
• Express the solution set in parametric form.
• Compute the rank of a 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. Our solutions will be in parametric form. For example, the solution to Example SAA in parametric form is $${\bf x} \ = \ \left[ \begin{array}{c} 3-x_3 \\ 2+x_3 \\ x_3 \\ \end{array} \right] \ = \ \left[\begin{array}{c} 3 \\ 2 \\ 0 \\ \end{array}\right] + x_3 \, \left[\begin{array}{c} -1 \\ 1 \\ 1 \\ \end{array}\right].$$
  • The following is a system of six equations in nine variables. Read through until they give the parametric form of the solution: http://linear.ups.edu/html/archetype-J.html.
• Homework:
  • From http://linear.ups.edu/html/section-SSLE.html, do Exercises SSLE.C30, SSLE.C31, SSLE.C32, SSLE.C33, and SSLE.C34. Put your solutions in parametric form.
  • 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. Put your solutions in parametric form.
    The parametric form of the solution for several of the above Exercises is given below:

    RREF.C12	RREF.C14	RREF.C15
    $${\bf x} \ = \ \left[\begin{array}{cc} 2 \cr 0 \cr -6 \cr 0 \end{array}\right] + x_2 \, \left[\begin{array}{cc} -2 \cr 1 \cr 0 \cr 0 \end{array}\right] + x_4 \, \left[\begin{array}{cc} 4 \cr 0 \cr -3 \cr 1 \end{array}\right]$$	$${\bf x} \ = \ \left[\begin{array}{cc} 3 \cr -2 \cr 0 \cr 0 \end{array}\right] + x_3 \, \left[\begin{array}{cc} -2 \cr -3 \cr 1 \cr 0 \end{array}\right] + x_4 \, \left[\begin{array}{cc} -1 \cr 4 \cr 0 \cr 1 \end{array}\right]$$	$${\bf x} \ = \ \left[\begin{array}{cc} 3 \cr -5 \cr 7 \cr 0 \end{array}\right] + x_4 \, \left[\begin{array}{cc} -2 \cr 3 \cr -4 \cr 1 \end{array}\right]$$

  • 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. Put your solutions in parametric 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. Homogeneous Systems

• Identify a homogeneous system.
• Use RREF$(A)$ to solve a homogeneous system.
• Express the solution set in parametric form.

• 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. Put your solutions in parametric form.
  • Do LA2suppHW1.pdf.