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