Linear dependence of a set of vectors

Example, Section 8 Chapter 3 of Boas

a1 = {1, 4, -5}; a2 = {5, 2, 1}; a3 = {2, -1, 3}; 
    a4 = {3, -6, 11};

Our criterion for linear dependence is that there be only the solutions a=b=c=d=0 if our 4 vectors are linearly independent.  Each non-zero solution eliminates 1 vector from the set.

Solve[
      a a1  + b a2  + c a3  + d a4 == 0, {a, b, c, 
        d}]

Conclude:the last TWO vectors are linearly independent since the first two can be expressed in terms of them.

Solve[
      a a1  + b a2  + c a3  + d a4 == 0, {c, d}]

Conclude:the FIRST TWO vectors are linearly independent since the last two can be expressed in terms of them.  Thus WHICH TWO are linearly independent.

Boas identifies the rank of the reduced matrix derived from the matrix whowe rows are the vectors as the number of linearly independent vectors.

mat = {a1, a2, a3, a4}; RowReduce[mat] // MatrixForm

matAlt = {a3, a1, a4, a2}; RowReduce[matAlt] // MatrixForm

This is not surprising: the order of the equations has no effect on the reduced matrix.