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  • MAT223 Readings 01

Leading and Non-leading Variables in Linear Algebra

• A leading 1, is a leading variable. Any variable not equal to $1$ and is not leading is a free / non-leading variable.
• Non-leading variables can be parameters we can change, when solving.
• Ex:
  • $R=\left[\begin{array}{cccccc}1& 0& 0& 1& -2& 2\\ 0& 0& 1& 5& 6& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]$
  • Leading constants are $a_{11}$ and $a_{23}$ then we have leading variables of $x_1$ and $x_3$
  • Non-leading is then $x_2,x_4,x_5$
  • On the top we have:
    • $x_1+x_4-2x_5=2$
    • $x_1=2-x_4+2x_5$
  • Second row:
    • $x_3+5x_4+6x_5=1$
    • $x_3=1-5x_4-6x_5$
  • Assign free variables:
    • $x_2=s$
    • $x_4=t$
    • $x_5=u$
  • Now our system is:
    • $\begin{array}{c}{x}_1=2-t+2u\\ x_3=1-5t-6u\end{array}$
    • So change the parameters: $s,t,u$ and you'll get every solution.
  • $x_1=2-t+2u$
  • $x_2=s$
  • $x_3=1-5t-6u$
  • $x_4=t$
  • $x_5=u$
  • Soln 1:
    • Let $(s,t,u)=(0,0,0)$
    • $(2,0,1,0,0)$
  • Soln 2:
    • Let $(s,t,u)=(1,1,1)$
    • $(3,1,-10,1,1)$
• Ex:
  • $\left[\begin{array}{ccccc}1& 0& 1& -2& 5\\ 0& 1& -2& 3& 1\\ 0& 0& 0& 0& 0\end{array}\right]$
  • $x_1+x_3-2x_4=5$
  • $x_2-2x_3+3x_4=1$
  • $x_1=5-x_3+2x_4$
  • $x_2=1+2x_3-3x_4$
  • Let $(x_3,x_4)=(s,t)$
  • $x_1=5-s+2t$
  • $x_2=1+2s-3t$