Hello,I am not confident in this Linear Algebra question: Suppose that a nonhomogeneous system with 10 linear equations in 8 unknowns has a solution with 2 free variables. Is it possible to change some constants on the equations’ right hand side to make the new system inconsistent? Explain in detail. Help would be greatly appreciated

Question
Asked Jul 26, 2019

Hello,

I am not confident in this Linear Algebra question:

 

Suppose that a nonhomogeneous system with 10 linear equations in 8 unknowns has a solution with 2 free variables. Is it possible to change some constants on the equations’ right hand side to make the new system inconsistent? Explain in detail.

 

Help would be greatly appreciated

 

 

check_circleExpert Solution
Step 1

Consider the system,

Ax b, where A is a 10x8 matrix
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Ax b, where A is a 10x8 matrix

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Step 2

It is given that two free variable.

dim(Nul A) nullity (A)
2 free variable]
=2
The rank theorem states, the dimension of the column space and the row space
of mxn matrix A are equal. This common dimension, the rank of A equals to
the number of pivot position in A and it satisfies rankA+dim(NulA)=n
That is, dim(NulA)=n-rankA
RankA 8-dim(Nul A)
8-2
= 6
help_outline

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dim(Nul A) nullity (A) 2 free variable] =2 The rank theorem states, the dimension of the column space and the row space of mxn matrix A are equal. This common dimension, the rank of A equals to the number of pivot position in A and it satisfies rankA+dim(NulA)=n That is, dim(NulA)=n-rankA RankA 8-dim(Nul A) 8-2 = 6

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Step 3

Thus, the dimension of column...

dim (Col A) rankA
- 6
Since ColA is a subspace of R, it can be concluded that Col A = 6
(R").
6
10
This implies that x-> Ax is onto R°
Hence, the new system with linear equations is inconsistent
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dim (Col A) rankA - 6 Since ColA is a subspace of R, it can be concluded that Col A = 6 (R"). 6 10 This implies that x-> Ax is onto R° Hence, the new system with linear equations is inconsistent

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