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?

Question
Equations
asked 2021-03-04
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?

Answers (1)

2021-03-05
Step 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 \(\displaystyle{A}={5}\times{6}\) matrix.
dim(NulA)=1 because all solutions of
Ax = 0 are multiples of one nonzero solution.
Step 2
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 \(\displaystyle{5}\times{6}\) matrix. n = 6
rank(A)=n-dim(NulA)=6-1=5
Image of A is 5-dimensional subspace of \(\displaystyle{R}^{{5}}\)(because A has 5 rows).
So, \(\displaystyle{C}{o}{l}{\left({A}\right)}={R}^{{5}}\)
This means that Ax=b has a solution for every b.
0

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