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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Answer 1

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)=n-dim(NulA)=6-1=5
Image of A is 5 dimensional subspace of \(R^5\) (because A has 5 rows)
So,Col(A)=\(R^5\)
This means that Ax=b has a solution for every b

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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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