Other examples.

• LU decomposition example:
  
  
  $$A=\left(\begin{array}{cccc} 1 & 2 \\ 3 & 4 \end{array}\right) \,$$
  
  
  LU2 returns:
  
  $$L^{-1}=\left(\begin{array}{rlc} 1 &0 \\ -3 & 1 \end{array}... ...U= \left(\begin{array}{rlc} 1 & 2\\ 0 & -2 \end{array}\right)$$
  
  
  We have A=LU.

• Rank of a matrix:
  
  
  $$\left(\begin{array}{cccc} 1 & 2& 4 & 6\\ -1 & 3 & 5 & 7 \\ 2 & 1 & 0 & 1 \\ 2 & 6 & 9 & 14 \end{array}\right)\ ,$$
  
  
  hit RANG, and look at the matrix:
  
  $$\left(\begin{array}{cccc} 1 &2 & 4 & 6 \\ 0 & 5 & 9 & 13 \\ 0 & 0 & -13 & -16\\ 0 & 0 &0 & 0 \end{array}\right)$$
  
  
  the rank is 3, the number of non zero lines (BTW you get a half reduced matrix)

• Linear relations between vectors
  Suppose we want to know the rank and linear relations existing between v₁(1,2,0), v₂(-2,-1,1), $v_{3}(0,3,1) \in I\!\!R ^3$:

  { { 1 2 0 V1 }
    { -2 -1 1 V2 }
    { 0 3 1 V3 } }
  

  then RANG, we get:

  { { 1 2 0 V1 }
    { 0 3 1 '2*V1+V2' }
    { 0 0 0 '-(2*V1)-V2+V3' } }
  

  The family is of rank 2 (the 3rd line is 0) and -2v₁-v₂+v₃=0.