Question

Is it possible that all solutions of a homogeneous system of fifteen linear equations in nineteen variables are multiples of one fixed nonzero​ solution? Discuss.

Answer 1

No, it is not possible

Explanation:

This is possible if the dimension of nullity of matrix A is unity.

That is, dim Nul A = 1

for A = 15 × 19.

The system has 19 variables, and by Rank Theorem, we have that

rank A + dim Nul A = 15 .

Since the rank can be at most 19, the null space has to have dimension at least 4.