Is it possible that all solutions of a homogeneous system of

Is it possible that all solutions of a homogeneous system of fourteen linear equations in nineteen variables are multiples of one fixed nonzero solution? Explain your answer.

Solution

Solution :

In mathematical language, the question asks whether it is possible for a 14 × 19 matrix A to have dim Nul A = 1 . The answer is no, because the Rank Theorem tells us that rank A + dim Nul A = 19 . Since the rank can be at most 14, the null space has to have dimension at least five.