: Suppose the field we're working with. You have to find out whether the coefficients HAVE TO BE zero or else the above equation is possible with at least one coefficient different from zero:
Now bring this matrix's coefficients into echelon form (reduce or prune the matrix using Gauss or Gauss-Jordan method) and you'll get that no row becomes all zeroes, which means the only possible solution to the above hmogeneous linear sistem is the trivial solution, i.e. and thus the vectors are lin. ind.