Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution

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Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution

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Aradjudyid Aradjudyid · Answer 1

Aradjudyid Aradjudyid

The answer is no, because the Rank Theorem tells us that rank A + dim Nul A = 12 . Since the rank can be at most 10, the null space has to have dimension at least two.

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