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.

Question

Mathematics

Trystan Baldwin

6 February, 07:57

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.

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Answers (1)

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Dallas Hurley · Answer 1

No, it is not possible

Step-by-step 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.