Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.

UkusakazaL 2020-12-24 Answered
Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.
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Willie
Answered 2020-12-25 Author has 95 answers

Consider the provided question,
Given a system of five linear equations in six unknowns are all multiples of one nonzero solution.
It can be written as,
Ax=0 where A=5×6 matrix
dim(NulA)=1 because all solutions of Ax=0 are multiples of nonzero solution
We have to explain that the system necessarily have a solution for every possible choice of constants on the right sides of the equations.
Since,A is 5×6 matrix n=6
rank(A)=ndim(NulA)=61=5
Image of A is 5 dimensional subspace of R5 (because A has 5 rows)
So,Col(A)=R5
This means that Ax=b has a solution for every b

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