Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent?

Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent?

Question
Equations
asked 2020-12-15
Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent?

Answers (1)

2020-12-16
Step 1
According to the given information, a non-homogeneous system of six linear equations in eight unknown has a solution with two free variables.
It is required to check whether is it possible to change some constants on the equations on right side to make new system inconsistence.
dim(Nul(A)) is same as number of free variables
so, dim(Nul(A))=2
Step 2
Since, n = 8 as system has eight unknowns so, rank of coefficient matrix is:
rank(A)=8-2=6
Step 3
Further, col(A) is a six-dimensional subspace of \(\displaystyle{R}^{{6}}\) because it has six linear equations so, \(\displaystyle{c}{o}{l}{\left({A}\right)}={R}^{{6}}\)
This means that Ax = b has a solution for every b and cannot change constant on right side to make system inconsistence.
Hence, it is not possible to change the constants.
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