A scientist has found two solutions to a homogeneous system of 40 equations in 42 variables. The two solutions are not multiples, and all other solutions can be constructed by adding together appropriate multiples of these two solutions. Can the scientist be certain that an associated

Question

A scientist has found two solutions to a homogeneous system of 40 equations in 42 variables. The two solutions are not multiples, and all other solutions can be constructed by adding together appropriate multiples of these two solutions. Can the scientist be certain that an associated nonhomogeneous system (with the same coefficients) has a solution?

Accepted Answer

Yes. Let A be the 40 × 42 coefficient matrix of the system. The given information implies that the two solutions are linearly independent and span Nul A. So dim Nul A = 2. By the Rank Theorem, dim Col A = 42 - 2 = 40. Since [latex] \mathbb{R} ^{40} [/latex] is the only subspace of [latex] \mathbb{R} ^{40} [/latex] whose dimension is 40, Col A must be all of [latex] \mathbb{R} ^{40} [/latex]. This means that every nonhomogeneous equation Ax = b has a solution.

Question 4.6.5: A scientist has found two solutions to a homogeneous system ...

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