Question

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.

Forms of linear equations
ANSWERED
asked 2020-12-24
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.

Answers (1)

2020-12-25

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 \(A=5\times6\) 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 \(5\times6\) 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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