Example 3

Let's look at something else. Let's look again at the GJSF of [d,f].

In[16]:=

  MatrixForm[RowReduce[df]]

Out[16]=

  1    0    0    -1   -2
  
  0    1    0    0    -1
  
  0    0    1    2    8

Notice that the last column in the GJSF of [d,f] is not a pivot column. That says that the last column is a linear combination of previous columns (recall the conditions on GJSF). If we write the new system this represents as d'x=f', this is precisely what it means for d'x=f' to have a solution! Remember, matrix multiplication is just taking linear combinations of columns, so there exists a linear combination of the columns of d' that yield f'.

If the last column is a pivot column, then f' cannot be a linear combination of previous columns, so d'x=f', hence dx=f, has no solution. Take the following for example.

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